Leveraged ETF Return Dispersion

Table of Contents

Introduction #

Over the past 15 years (or so), leveraged ETFs have become frequently used for trading equity indices, sectors, and other asset classes by the investor that is seeking to use leverage for excess exposure to those asset classes. The question remains, however, what happens to the returns of leveraged ETFs over an extended time horizon and is there an optimal leverage ratio for the long-term buy-and-hold investor that allows them to take advantage of leverage to increase the up-side returns, while avoiding catastrophic losses on the down-side? In this investigation, we will delve into these ideas and see what the data shows.

Python Imports #

# Standard Library
import os
import sys
import warnings

from pathlib import Path

# Data Handling
import pandas as pd

# Suppress warnings
warnings.filterwarnings("ignore")

# Add the source subdirectory to the system path to allow import config from settings.py
current_directory = Path(os.getcwd())
website_base_directory = current_directory.parent.parent.parent
src_directory = website_base_directory / "src"
sys.path.append(str(src_directory)) if str(src_directory) not in sys.path else None

# Import settings.py
from settings import config

# Add configured directories from config to path
SOURCE_DIR = config("SOURCE_DIR")
sys.path.append(str(Path(SOURCE_DIR))) if str(Path(SOURCE_DIR)) not in sys.path else None

# Add other configured directories
BASE_DIR = config("BASE_DIR")
CONTENT_DIR = config("CONTENT_DIR")
POSTS_DIR = config("POSTS_DIR")
PAGES_DIR = config("PAGES_DIR")
PUBLIC_DIR = config("PUBLIC_DIR")
SOURCE_DIR = config("SOURCE_DIR")
DATA_DIR = config("DATA_DIR")
DATA_MANUAL_DIR = config("DATA_MANUAL_DIR")

Python Functions #

Here are the functions needed for this project:

load_data: Load data from a CSV, Excel, or Pickle file into a pandas DataFrame.
pandas_set_decimal_places: Set the number of decimal places displayed for floating-point numbers in pandas.
plot_histogram: Plot the histogram of a data set from a DataFrame.
plot_scatter: Plot the data from a DataFrame for a specified date range and columns.
plot_time_series: Plot the timeseries data from a DataFrame for a specified date range and columns.
run_regression: Run a linear regression using statsmodels OLS and return the results.
summary_stats: Generate summary statistics for a series of returns.
yf_pull_data: Download daily price data from Yahoo Finance and export it.

from load_data import load_data
from pandas_set_decimal_places import pandas_set_decimal_places
from plot_histogram import plot_histogram
from plot_scatter import plot_scatter
from plot_time_series import plot_time_series
from run_regression import run_regression
from summary_stats import summary_stats
from yf_pull_data import yf_pull_data

Data Overview #

For this exercise, we will investigate the long-term return relationships between the following:

QQQ (Invesco QQQ Trust, Series 1) and TQQQ (ProShares UltraPro QQQ)
SPY (SPDR S&P 500 ETF Trust) and UPRO (ProShares UltraPro S&P 500)

Just to clarify, any time we are referring to “close prices” in this analysis, we are referring to the partially-adjusted close prices that account for splits, but not dividends. Because we are dealing with leveraged ETFs, we want to focus on the pure returns due to change in price, but exclude the dividends, which are not leveraged in the same way as the price changes.

QQQ & TQQQ #

Acquire & Plot Data (QQQ) #

First, let’s get the data for QQQ. If we already have the desired data, we can load it from a local pickle file. Otherwise, we can download it from Yahoo Finance using the yf_pull_data function.

pandas_set_decimal_places(2)

yf_pull_data(
    base_directory=DATA_DIR,
    ticker="QQQ",
    adjusted=False,
    source="Yahoo_Finance",
    asset_class="Exchange_Traded_Funds",
    excel_export=True,
    pickle_export=True,
    output_confirmation=False,
)

qqq = load_data(
    base_directory=DATA_DIR,
    ticker="QQQ",
    source="Yahoo_Finance",
    asset_class="Exchange_Traded_Funds",
    timeframe="Daily",
    file_format="pickle",
)

# Rename columns to "QQQ_Close", etc.
qqq = qqq.rename(columns={
    "Adj Close": "QQQ_Adj_Close",
    "Close": "QQQ_Close",
    "High": "QQQ_High",
    "Low": "QQQ_Low",
    "Open": "QQQ_Open",
    "Volume": "QQQ_Volume"
})

display(qqq)

	QQQ_Adj_Close	QQQ_Close	QQQ_High	QQQ_Low	QQQ_Open	QQQ_Volume
Date
1999-03-10	43.07	51.06	51.16	50.28	51.12	5232000
1999-03-11	43.29	51.31	51.73	50.31	51.44	9688600
1999-03-12	42.23	50.06	51.16	49.66	51.12	8743600
1999-03-15	43.44	51.50	51.56	49.91	50.44	6369000
1999-03-16	43.81	51.94	52.16	51.16	51.72	4905800
...	...	...	...	...	...	...
2026-06-10	693.69	693.69	711.28	692.93	701.66	65334300
2026-06-11	717.12	717.12	718.37	695.00	699.29	71798900
2026-06-12	721.34	721.34	724.01	711.28	717.61	51168400
2026-06-15	744.00	744.00	744.76	737.38	738.10	46710200
2026-06-16	729.86	729.86	744.22	729.64	742.25	45348700

6860 rows × 6 columns

And the plot of the time series of partially adjusted close prices:

plot_time_series(
    df=qqq,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Adj_Close"],
    title="QQQ Adjusted Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

Acquire & Plot Data (TQQQ) #

Next, TQQQ:

yf_pull_data(
    base_directory=DATA_DIR,
    ticker="TQQQ",
    adjusted=False,
    source="Yahoo_Finance", 
    asset_class="Exchange_Traded_Funds", 
    excel_export=True,
    pickle_export=True,
    output_confirmation=False,
)

tqqq = load_data(
    base_directory=DATA_DIR,
    ticker="TQQQ",
    source="Yahoo_Finance", 
    asset_class="Exchange_Traded_Funds",
    timeframe="Daily",
    file_format="pickle",
)

# Rename columns to "TQQQ_Close", etc.
tqqq = tqqq.rename(columns={
    "Adj Close": "TQQQ_Adj_Close",
    "Close": "TQQQ_Close", 
    "High": "TQQQ_High", 
    "Low": "TQQQ_Low", 
    "Open": "TQQQ_Open", 
    "Volume": "TQQQ_Volume"
})

display(tqqq)

	TQQQ_Adj_Close	TQQQ_Close	TQQQ_High	TQQQ_Low	TQQQ_Open	TQQQ_Volume
Date
2010-02-11	0.21	0.22	0.22	0.20	0.20	6912000
2010-02-12	0.21	0.22	0.22	0.21	0.21	17203200
2010-02-16	0.21	0.23	0.23	0.22	0.22	19238400
2010-02-17	0.22	0.23	0.23	0.23	0.23	38361600
2010-02-18	0.22	0.23	0.24	0.23	0.23	77721600
...	...	...	...	...	...	...
2026-06-10	69.27	69.27	74.70	69.00	71.68	91465200
2026-06-11	76.01	76.01	76.60	69.59	70.89	116288800
2026-06-12	77.52	77.52	78.36	74.29	76.34	94070700
2026-06-15	84.59	84.59	85.03	82.64	82.88	59758000
2026-06-16	79.93	79.93	84.83	79.86	84.19	67086700

4111 rows × 6 columns

And the plot of the time series of partially adjusted close prices:

plot_time_series(
    df=tqqq,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["TQQQ_Adj_Close"],
    title="TQQQ Adjusted Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

Looking at the close prices doesn’t give us a true picture of the magnitude of the difference in returns due to the leverage. In order to see that, we need to look at the cumulative returns and the drawdowns.

Calculate & Plot Cumulative Returns, Rolling Returns, and Drawdowns (QQQ & TQQQ) #

Next, we will calculate the cumulative returns, rolling returns, and drawdowns. This involves aligning the data to start with the inception of TQQQ. For this excercise, we will not extrapolate the data for QQQ back to 1999, but rather just align the data from the inception of TQQQ in 2010.

etfs = ["QQQ", "TQQQ"]

# Merge dataframes and drop rows with missing values
qqq_tqqq_aligned = tqqq.merge(qqq, left_index=True, right_index=True, how='left')
qqq_tqqq_aligned = qqq_tqqq_aligned.dropna()

# Calculate cumulative returns
for etf in etfs:
    qqq_tqqq_aligned[f"{etf}_Return"] = qqq_tqqq_aligned[f"{etf}_Close"].pct_change()
    qqq_tqqq_aligned[f"{etf}_Cumulative_Return"] = (1 + qqq_tqqq_aligned[f"{etf}_Return"]).cumprod() - 1
    qqq_tqqq_aligned[f"{etf}_Cumulative_Return_Plus_One"] = 1 + qqq_tqqq_aligned[f"{etf}_Cumulative_Return"]
    qqq_tqqq_aligned[f"{etf}_Rolling_Max"] = qqq_tqqq_aligned[f"{etf}_Cumulative_Return_Plus_One"].cummax()
    qqq_tqqq_aligned[f"{etf}_Drawdown"] = qqq_tqqq_aligned[f"{etf}_Cumulative_Return_Plus_One"] / qqq_tqqq_aligned[f"{etf}_Rolling_Max"] - 1
    qqq_tqqq_aligned.drop(columns=[f"{etf}_Cumulative_Return_Plus_One", f"{etf}_Rolling_Max"], inplace=True)

# Define rolling windows in trading days
rolling_windows = {
    '1d': 1,      # 1 day
    '1w': 5,      # 1 week (5 trading days)
    '1m': 21,     # 1 month (~21 trading days)
    '3m': 63,     # 3 months (~63 trading days)
    '6m': 126,    # 6 months (~126 trading days)
    '1y': 252,    # 1 year (~252 trading days)
    '2y': 504,    # 2 years (~504 trading days)
    '3y': 756,    # 3 years (~756 trading days)
    '4y': 1008,   # 4 years (~1008 trading days)
    '5y': 1260,   # 5 years (~1260 trading days)
}

# Calculate rolling returns for each ETF and each window
for etf in etfs:
    for period_name, window in rolling_windows.items():
        qqq_tqqq_aligned[f"{etf}_Rolling_Return_{period_name}"] = (
            qqq_tqqq_aligned[f"{etf}_Close"].pct_change(periods=window)
        )
        
display(qqq_tqqq_aligned)

	TQQQ_Adj_Close	TQQQ_Close	TQQQ_High	TQQQ_Low	TQQQ_Open	TQQQ_Volume	QQQ_Adj_Close	QQQ_Close	QQQ_High	QQQ_Low	...	TQQQ_Rolling_Return_1d	TQQQ_Rolling_Return_1w	TQQQ_Rolling_Return_1m	TQQQ_Rolling_Return_3m	TQQQ_Rolling_Return_6m	TQQQ_Rolling_Return_1y	TQQQ_Rolling_Return_2y	TQQQ_Rolling_Return_3y	TQQQ_Rolling_Return_4y	TQQQ_Rolling_Return_5y
Date
2010-02-11	0.21	0.22	0.22	0.20	0.20	6912000	37.90	43.67	43.79	42.76	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2010-02-12	0.21	0.22	0.22	0.21	0.21	17203200	37.98	43.76	43.88	43.16	...	0.00	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2010-02-16	0.21	0.23	0.23	0.22	0.22	19238400	38.47	44.32	44.35	43.85	...	0.04	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2010-02-17	0.22	0.23	0.23	0.23	0.23	38361600	38.69	44.57	44.57	44.26	...	0.02	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2010-02-18	0.22	0.23	0.24	0.23	0.23	77721600	38.93	44.85	44.93	44.45	...	0.02	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2026-06-10	69.27	69.27	74.70	69.00	71.68	91465200	693.69	693.69	711.28	692.93	...	-0.06	-0.20	-0.10	0.40	0.24	0.86	1.05	2.75	2.97	1.80
2026-06-11	76.01	76.01	76.60	69.59	70.89	116288800	717.12	717.12	718.37	695.00	...	0.10	-0.11	0.01	0.62	0.36	1.00	1.26	3.12	3.73	1.92
2026-06-12	77.52	77.52	78.36	74.29	76.34	94070700	721.34	721.34	724.01	711.28	...	0.02	0.06	0.00	0.69	0.37	1.06	1.31	3.43	3.76	1.96
2026-06-15	84.59	84.59	85.03	82.64	82.88	59758000	744.00	744.00	744.76	737.38	...	0.09	0.11	0.07	0.78	0.51	1.24	1.49	3.68	4.08	2.22
2026-06-16	79.93	79.93	84.83	79.86	84.19	67086700	729.86	729.86	744.22	729.64	...	-0.06	0.08	0.06	0.66	0.51	1.19	1.31	3.37	3.90	2.04

4111 rows × 38 columns

And now the plot for the cumulative returns:

plot_time_series(
    df=qqq_tqqq_aligned,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Cumulative_Return", "TQQQ_Cumulative_Return"],
    title="Cumulative Returns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Cumulative Return",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

And the drawdown plot:

plot_time_series(
    df=qqq_tqqq_aligned,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Drawdown", "TQQQ_Drawdown"],
    title="Drawdowns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Drawdown",
    y_format="Percentage",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

Here is where we truly see the volatility of TQQQ relative to QQQ. In the past 5 years, TQQQ has had drawdowns of 50%, 60%, 70%, and 80%. While it has recovered to make new highs (with the exception of the current ~25% drawdown as of mid-March 2026), very few investors can endure those drawdowns and continue to hold their position. At the same time, we can see from the plot that a ~35% drawdown in QQQ equated to a ~80% drawdown in TQQQ, which is not in fact, 3x. So this tells us (which we already knew) that there is dispersion in the long-term returns relative to the short-term returns between the non-leveraged QQQ and 3x leveraged TQQQ. This idea is well documented in the financial literature as “volatility decay” or “volatility drag”. But, and this is the question we are trying to answer, how significant is this effect over various time horizons?

Summary Statistics (QQQ & TQQQ) #

Looking at the summary statistics further confirms our intuitions about the volatility and drawdowns.

qqq_sum_stats = summary_stats(
    fund_list=["QQQ"],
    df=qqq_tqqq_aligned[["QQQ_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

tqqq_sum_stats = summary_stats(
    fund_list=["TQQQ"],
    df=qqq_tqqq_aligned[["TQQQ_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

sum_stats = pd.concat([qqq_sum_stats, tqqq_sum_stats])

display(sum_stats)

	Annual Mean Return (Arithmetic)	Annualized Volatility	Annualized Sharpe Ratio	CAGR (Geometric)	Daily Max Return	Daily Max Return (Date)	Daily Min Return	Daily Min Return (Date)	Max Drawdown	Peak	Trough	Recovery Date	Calendar Days to Recovery	MAR Ratio
QQQ_Return	0.19	0.21	0.94	0.19	0.12	2025-04-09	-0.12	2020-03-16	-0.36	2021-11-19	2022-12-28	2023-12-15	352	0.53
TQQQ_Return	0.55	0.61	0.90	0.44	0.35	2025-04-09	-0.34	2020-03-16	-0.82	2021-11-19	2022-12-28	2024-12-11	714	0.53

Note that these statistics are being run on the partially-adjusted close prices, which are not the true returns (due to not accounting for dividends), but they do give us a picture of the relative volatility and drawdowns of the two ETFs. The mean return for TQQQ is much higher than that of QQQ, but the volatility is also much higher, which is consistent with the idea of leverage amplifying both the up-side and down-side. The maximum drawdown for TQQQ is also much higher than that of QQQ, which again confirms our observations from the drawdown plot.

Also note that the daily maximum return for both funds occured during “Liberation Day” and the daily minimum return for both funds occured early on during the COVID-19 pandemic.

Plot Returns & Verify Beta (QQQ & TQQQ) #

Before we look at the rolling returns, let us first verify that the daily returns for TQQQ are in fact ~3x those of QQQ. We can do that by plotting the daily returns for both funds against each other and running a linear regression to see if the beta is indeed ~3.

plot_scatter(
    df=qqq_tqqq_aligned,
    x_plot_column="QQQ_Return",
    y_plot_columns=["TQQQ_Return"],
    title="QQQ & TQQQ Returns",
    x_label="QQQ Return",
    x_format="Decimal",
    x_format_decimal_places=2,
    x_tick_spacing="Auto",
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="TQQQ Return",
    y_format="Decimal",
    y_format_decimal_places=2,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=True,
    OLS_column="TQQQ_Return",
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=True,
    RidgeCV_column="TQQQ_Return",
    regression_constant=True,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

model = run_regression(
    df=qqq_tqqq_aligned,
    x_plot_column="QQQ_Return",
    y_plot_column="TQQQ_Return",
    regression_model="OLS-statsmodels",
    regression_constant=True,
)

print(model.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:            TQQQ_Return   R-squared:                       0.997
Model:                            OLS   Adj. R-squared:                  0.997
Method:                 Least Squares   F-statistic:                 1.540e+06
Date:                Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                        14:31:49   Log-Likelihood:                 19741.
No. Observations:                4110   AIC:                        -3.948e+04
Df Residuals:                    4108   BIC:                        -3.946e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const      -8.798e-05    3.1e-05     -2.836      0.005      -0.000   -2.72e-05
QQQ_Return     2.9553      0.002   1240.806      0.000       2.951       2.960
==============================================================================
Omnibus:                     5361.302   Durbin-Watson:                   2.568
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          9467735.605
Skew:                          -6.352   Prob(JB):                         0.00
Kurtosis:                     237.786   Cond. No.                         76.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Visually, this plot makes sense and we can see that there is a strong clustering of points, but we double check with the regression, regressing the TQQQ daily return (y) on the QQQ daily return (X).

Given the above result, with a coefficient of 2.96 and an R^2 of 0.997 (based on the statsmodels OLS regression), we can say that TQQQ does in fact return ~3x QQQ. We would also intuitively expect the coefficient to be 0, and it is nearly 0.

Interestingly, the coefficient varies between OLS and Ridge cross-validation, and both are less than 3.

Extrapolate Data (QQQ & TQQQ) #

With the above coefficient, we will now extrapolate the returns of QQQ to backfill the data from the inception of QQQ in 1999 to the inception of TQQQ in 2010 to expand our dataset of returns. For this, we’ll use the coefficient of 2.96 that we found in the regression results above.

# Set leverage multiplier based on regression coefficient
LEVERAGE_MULTIPLIER = model.params.iloc[1]

# Merge dataframes and extrapolate return values for QQQ back to 1999 using the leverage multiplier
qqq_tqqq_extrap = qqq[["QQQ_Close"]].merge(tqqq[["TQQQ_Close"]], left_index=True, right_index=True, how='left')

etfs = ["QQQ", "TQQQ"]

# Calculate cumulative returns
for etf in etfs:
    qqq_tqqq_extrap[f"{etf}_Return"] = qqq_tqqq_extrap[f"{etf}_Close"].pct_change()

# Extrapolate TQQQ returns for missing values
qqq_tqqq_extrap["TQQQ_Return"] = qqq_tqqq_extrap["TQQQ_Return"].fillna(LEVERAGE_MULTIPLIER * qqq_tqqq_extrap["QQQ_Return"])

# Find the first valid TQQQ_Close index and value
first_valid_idx = qqq_tqqq_extrap['TQQQ_Close'].first_valid_index()
print(first_valid_idx)
first_valid_price = qqq_tqqq_extrap.loc[first_valid_idx, 'TQQQ_Close']
print(first_valid_price)

2010-02-11 00:00:00
0.21627600491046906

Before we extrapolate, let’s first look at the data we have for QQQ and TQQQ around the inception of TQQQ in 2010:

# Check values around the first valid index
pandas_set_decimal_places(4)
display(qqq_tqqq_extrap.loc["2010-02-08":"2010-02-13"])

	QQQ_Close	TQQQ_Close	QQQ_Return	TQQQ_Return
Date
2010-02-08	42.6700	NaN	-0.0072	-0.0213
2010-02-09	43.1100	NaN	0.0103	0.0305
2010-02-10	43.0200	NaN	-0.0021	-0.0062
2010-02-11	43.6700	0.2163	0.0151	0.0447
2010-02-12	43.7600	0.2172	0.0021	0.0041

Now, backfill the data for the TQQQ close price:

# Iterate through the dataframe backwards
for i in range(qqq_tqqq_extrap.index.get_loc(first_valid_idx) - 1, -1, -1):
    
    # The return that led to the price the next day
    current_return = qqq_tqqq_extrap.iloc[i + 1]['TQQQ_Return']

    # Get the next day's price
    next_price = qqq_tqqq_extrap.iloc[i + 1]['TQQQ_Close']
    
    # Price_{t} = Price_{t+1} / (1 + Return_{t})
    qqq_tqqq_extrap.loc[qqq_tqqq_extrap.index[i], 'TQQQ_Close'] = next_price / (1 + current_return)

Finally, confirm the values are correct:

# Confirm values around the first valid index after extrapolation
display(qqq_tqqq_extrap.loc["2010-02-08":"2010-02-13"])

	QQQ_Close	TQQQ_Close	QQQ_Return	TQQQ_Return
Date
2010-02-08	42.6700	0.2022	-0.0072	-0.0213
2010-02-09	43.1100	0.2083	0.0103	0.0305
2010-02-10	43.0200	0.2070	-0.0021	-0.0062
2010-02-11	43.6700	0.2163	0.0151	0.0447
2010-02-12	43.7600	0.2172	0.0021	0.0041

And the complete DataFrame with the extrapolated values:

pandas_set_decimal_places(2)
display(qqq_tqqq_extrap)

	QQQ_Close	TQQQ_Close	QQQ_Return	TQQQ_Return
Date
1999-03-10	51.06	13.82	NaN	NaN
1999-03-11	51.31	14.02	0.00	0.01
1999-03-12	50.06	13.01	-0.02	-0.07
1999-03-15	51.50	14.12	0.03	0.08
1999-03-16	51.94	14.47	0.01	0.03
...	...	...	...	...
2026-06-10	693.69	69.27	-0.02	-0.06
2026-06-11	717.12	76.01	0.03	0.10
2026-06-12	721.34	77.52	0.01	0.02
2026-06-15	744.00	84.59	0.03	0.09
2026-06-16	729.86	79.93	-0.02	-0.06

6860 rows × 4 columns

After the extrapolation, we now have the following plots for the prices, cumulative returns, and drawdowns:

etfs = ["QQQ", "TQQQ"]

# Calculate cumulative returns
for etf in etfs:
    qqq_tqqq_extrap[f"{etf}_Return"] = qqq_tqqq_extrap[f"{etf}_Close"].pct_change()
    qqq_tqqq_extrap[f"{etf}_Cumulative_Return"] = (1 + qqq_tqqq_extrap[f"{etf}_Return"]).cumprod() - 1
    qqq_tqqq_extrap[f"{etf}_Cumulative_Return_Plus_One"] = 1 + qqq_tqqq_extrap[f"{etf}_Cumulative_Return"]
    qqq_tqqq_extrap[f"{etf}_Rolling_Max"] = qqq_tqqq_extrap[f"{etf}_Cumulative_Return_Plus_One"].cummax()
    qqq_tqqq_extrap[f"{etf}_Drawdown"] = qqq_tqqq_extrap[f"{etf}_Cumulative_Return_Plus_One"] / qqq_tqqq_extrap[f"{etf}_Rolling_Max"] - 1
    qqq_tqqq_extrap.drop(columns=[f"{etf}_Cumulative_Return_Plus_One", f"{etf}_Rolling_Max"], inplace=True)

plot_time_series(
    df=qqq_tqqq_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Close"],
    title="QQQ Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=qqq_tqqq_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["TQQQ_Close"],
    title="TQQQ Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=qqq_tqqq_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Cumulative_Return", "TQQQ_Cumulative_Return"],
    title="Cumulative Returns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Cumulative Return",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=qqq_tqqq_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["QQQ_Drawdown", "TQQQ_Drawdown"],
    title="Drawdowns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Drawdown",
    y_format="Percentage",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

qqq_extrap_sum_stats = summary_stats(
    fund_list=["QQQ"],
    df=qqq_tqqq_extrap[["QQQ_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

tqqq_extrap_sum_stats = summary_stats(
    fund_list=["TQQQ"],
    df=qqq_tqqq_extrap[["TQQQ_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

sum_stats = pd.concat([qqq_sum_stats, tqqq_sum_stats, qqq_extrap_sum_stats, tqqq_extrap_sum_stats])
sum_stats.index = ["QQQ (2010 - Present)", "TQQQ (2010 - Present)", "QQQ (1999 - Present)", "TQQQ Extrapolated (1999 - Present)"]

display(sum_stats)

	Annual Mean Return (Arithmetic)	Annualized Volatility	Annualized Sharpe Ratio	CAGR (Geometric)	Daily Max Return	Daily Max Return (Date)	Daily Min Return	Daily Min Return (Date)	Max Drawdown	Peak	Trough	Recovery Date	Calendar Days to Recovery	MAR Ratio
QQQ (2010 - Present)	0.19	0.21	0.94	0.19	0.12	2025-04-09	-0.12	2020-03-16	-0.36	2021-11-19	2022-12-28	2023-12-15	352.00	0.53
TQQQ (2010 - Present)	0.55	0.61	0.90	0.44	0.35	2025-04-09	-0.34	2020-03-16	-0.82	2021-11-19	2022-12-28	2024-12-11	714.00	0.53
QQQ (1999 - Present)	0.13	0.27	0.50	0.10	0.17	2001-01-03	-0.12	2020-03-16	-0.83	2000-03-27	2002-10-09	2016-09-06	5081.00	0.12
TQQQ Extrapolated (1999 - Present)	0.38	0.80	0.48	0.07	0.50	2001-01-03	-0.34	2020-03-16	-1.00	2000-03-27	2009-03-09	NaT	NaN	0.07

A few quick comments before we look at rolling returns:

The cumulative return for TQQQ is less than that of QQQ - which is starkly different from the plot beginning in 2010 at the inception of TQQQ. So the return path really matters here.
The drawdown for TQQQ is nearly 100%… which also represents nearly a total loss of capital for any allocation to the extrap-TQQQ. Furthermore, as we walk forward through time (2002, 2003, … etc.), there is really no reason to believe that the returns would ever recover (even partially). So while we can look at the rolling returns and see how they compare to the 3x return of QQQ, we should keep in mind that the drawdown post-1999 is so severe that it would be very difficult for any investor to hold through it.
The recovery time for QQQ was more than 5,000 days, or ~14 years. Note that this is calendar days, not trading days. While returns have been great for QQQ since 2016, the 14 year dry spell is a reminder of just how large the tech bubble was.
The extrapolated TQQQ data remains in a drawdown and has never recovered to make new highs (as of March 2026).

Plot Rolling Returns (QQQ & TQQQ) #

Next, we will consider the following:

Histogram and scatter plots of the rolling returns of QQQ and TQQQ
Regressions to establish a “leverage factor” for the rolling returns
The deviation from a 3x return for each time period

For this set of regressions, we will also allow the constant. First, we need the rolling returns for various time periods:

# Define rolling windows in trading days
rolling_windows = {
    '1d': 1,      # 1 day
    '1w': 5,      # 1 week (5 trading days)
    '1m': 21,     # 1 month (~21 trading days)
    '3m': 63,     # 3 months (~63 trading days)
    '6m': 126,    # 6 months (~126 trading days)
    '1y': 252,    # 1 year (~252 trading days)
    '2y': 504,    # 2 years (~504 trading days)
    '3y': 756,    # 3 years (~756 trading days)
    '4y': 1008,   # 4 years (~1008 trading days)
    '5y': 1260,   # 5 years (~1260 trading days)
}

# Calculate rolling returns for each ETF and each window
for etf in etfs:
    for period_name, window in rolling_windows.items():
        qqq_tqqq_extrap[f"{etf}_Rolling_Return_{period_name}"] = (
            qqq_tqqq_extrap[f"{etf}_Close"].pct_change(periods=window)
        )

This gives us the following series of histograms, scatter plots, and regression model results:

# Create a dataframe to hold rolling returns stats
rolling_returns_stats = pd.DataFrame()

for period_name, window in rolling_windows.items():
    plot_histogram(
        df=qqq_tqqq_extrap,
        plot_columns=[f"QQQ_Rolling_Return_{period_name}", f"TQQQ_Rolling_Return_{period_name}"],
        title=f"QQQ & TQQQ {period_name} Rolling Returns",
        x_label="Rolling Return",
        x_tick_spacing="Auto",
        x_tick_rotation=30,
        y_label="# Of Datapoints",
        y_tick_spacing="Auto",
        y_tick_rotation=0,
        grid=True,
        legend=True,
        export_plot=False,
        plot_file_name=None,
    )

    plot_scatter(
        df=qqq_tqqq_extrap,
        x_plot_column=f"QQQ_Rolling_Return_{period_name}",
        y_plot_columns=[f"TQQQ_Rolling_Return_{period_name}"],
        title=f"QQQ & TQQQ {period_name} Rolling Returns",
        x_label="QQQ Rolling Return",
        x_format="Decimal",
        x_format_decimal_places=2,
        x_tick_spacing="Auto",
        x_tick_start=None,
        x_tick_rotation=30,
        y_label="TQQQ Rolling Return",
        y_format="Decimal",
        y_format_decimal_places=2,
        y_tick_spacing="Auto",
        y_tick_rotation=0,
        plot_OLS_regression_line=True,
        OLS_column=f"TQQQ_Rolling_Return_{period_name}",
        plot_Ridge_regression_line=False,
        Ridge_column=None,
        plot_RidgeCV_regression_line=True,
        RidgeCV_column=f"TQQQ_Rolling_Return_{period_name}",
        regression_constant=True,
        grid=True,
        legend=True,
        export_plot=False,
        plot_file_name=None,
    )

    # Run OLS regression with statsmodels
    model = run_regression(
        df=qqq_tqqq_extrap,
        x_plot_column=f"QQQ_Rolling_Return_{period_name}",
        y_plot_column=f"TQQQ_Rolling_Return_{period_name}",
        regression_model="OLS-statsmodels",
        regression_constant=True,
    )
    print(model.summary())

    # Add the regression results to the rolling returns stats dataframe
    intercept = model.params.iloc[0]
    intercept_pvalue = model.pvalues.iloc[0]   # p-value for Intercept
    slope = model.params.iloc[1]
    slope_pvalue = model.pvalues.iloc[1]       # p-value for QQQ_Return
    r_squared = model.rsquared

    # Calc skew
    return_ratio = qqq_tqqq_extrap[f'TQQQ_Rolling_Return_{period_name}'] / qqq_tqqq_extrap[f'QQQ_Rolling_Return_{period_name}']
    skew = return_ratio.skew()

    # Calc conditional symmetry
    up_markets = qqq_tqqq_extrap[qqq_tqqq_extrap[f'QQQ_Rolling_Return_{period_name}'] > 0]
    down_markets = qqq_tqqq_extrap[qqq_tqqq_extrap[f'QQQ_Rolling_Return_{period_name}'] <= 0]

    avg_beta_up = (up_markets[f'TQQQ_Rolling_Return_{period_name}'] / up_markets[f'QQQ_Rolling_Return_{period_name}']).mean()
    avg_beta_down = (down_markets[f'TQQQ_Rolling_Return_{period_name}'] / down_markets[f'QQQ_Rolling_Return_{period_name}']).mean()

    asymmetry = avg_beta_up - avg_beta_down

    rolling_returns_slope_int = pd.DataFrame({
        "Period": period_name,
        "Intercept": [intercept],
        # "Intercept_PValue": [intercept_pvalue],
        "Slope": [slope],
        # "Slope_PValue": [slope_pvalue],
        "R_Squared": [r_squared],
        "Return Skew": [skew],
        "Average Upside Beta": [avg_beta_up],
        "Average Downside Beta": [avg_beta_down],
        "Asymmetry": [asymmetry]
    })

    rolling_returns_stats = pd.concat([rolling_returns_stats, rolling_returns_slope_int])

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_1d   R-squared:                       0.999
Model:                                OLS   Adj. R-squared:                  0.999
Method:                     Least Squares   F-statistic:                 7.303e+06
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:31:57   Log-Likelihood:                 34698.
No. Observations:                    6859   AIC:                        -6.939e+04
Df Residuals:                        6857   BIC:                        -6.938e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                  -5.27e-05   1.86e-05     -2.837      0.005   -8.91e-05   -1.63e-05
QQQ_Rolling_Return_1d     2.9553      0.001   2702.435      0.000       2.953       2.957
==============================================================================
Omnibus:                    10277.277   Durbin-Watson:                   2.566
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         44417091.716
Skew:                          -8.263   Prob(JB):                         0.00
Kurtosis:                     396.884   Cond. No.                         58.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_1w   R-squared:                       0.994
Model:                                OLS   Adj. R-squared:                  0.994
Method:                     Least Squares   F-statistic:                 1.134e+06
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:01   Log-Likelihood:                 23396.
No. Observations:                    6855   AIC:                        -4.679e+04
Df Residuals:                        6853   BIC:                        -4.677e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0008   9.65e-05     -8.403      0.000      -0.001      -0.001
QQQ_Rolling_Return_1w     2.9532      0.003   1064.980      0.000       2.948       2.959
==============================================================================
Omnibus:                     2863.007   Durbin-Watson:                   0.931
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           576171.657
Skew:                          -0.858   Prob(JB):                         0.00
Kurtosis:                      47.881   Cond. No.                         28.8
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_1m   R-squared:                       0.982
Model:                                OLS   Adj. R-squared:                  0.982
Method:                     Least Squares   F-statistic:                 3.747e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:05   Log-Likelihood:                 14971.
No. Observations:                    6839   AIC:                        -2.994e+04
Df Residuals:                        6837   BIC:                        -2.992e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0036      0.000    -10.803      0.000      -0.004      -0.003
QQQ_Rolling_Return_1m     2.9365      0.005    612.153      0.000       2.927       2.946
==============================================================================
Omnibus:                     1652.780   Durbin-Watson:                   0.293
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            67847.108
Skew:                           0.383   Prob(JB):                         0.00
Kurtosis:                      18.411   Cond. No.                         14.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_3m   R-squared:                       0.959
Model:                                OLS   Adj. R-squared:                  0.959
Method:                     Least Squares   F-statistic:                 1.577e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:09   Log-Likelihood:                 8045.5
No. Observations:                    6797   AIC:                        -1.609e+04
Df Residuals:                        6795   BIC:                        -1.607e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0083      0.001     -8.979      0.000      -0.010      -0.007
QQQ_Rolling_Return_3m     2.9871      0.008    397.059      0.000       2.972       3.002
==============================================================================
Omnibus:                     3498.049   Durbin-Watson:                   0.105
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            80662.481
Skew:                           1.963   Prob(JB):                         0.00
Kurtosis:                      19.414   Cond. No.                         8.38
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_6m   R-squared:                       0.916
Model:                                OLS   Adj. R-squared:                  0.916
Method:                     Least Squares   F-statistic:                 7.321e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:12   Log-Likelihood:                 2661.5
No. Observations:                    6734   AIC:                            -5319.
Df Residuals:                        6732   BIC:                            -5305.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0102      0.002     -4.853      0.000      -0.014      -0.006
QQQ_Rolling_Return_6m     3.0396      0.011    270.569      0.000       3.018       3.062
==============================================================================
Omnibus:                     3714.404   Durbin-Watson:                   0.056
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            61899.283
Skew:                           2.278   Prob(JB):                         0.00
Kurtosis:                      17.137   Cond. No.                         5.68
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_1y   R-squared:                       0.881
Model:                                OLS   Adj. R-squared:                  0.881
Method:                     Least Squares   F-statistic:                 4.869e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:16   Log-Likelihood:                -883.92
No. Observations:                    6608   AIC:                             1772.
Df Residuals:                        6606   BIC:                             1785.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                     0.0191      0.004      5.072      0.000       0.012       0.027
QQQ_Rolling_Return_1y     2.8417      0.013    220.652      0.000       2.816       2.867
==============================================================================
Omnibus:                     3502.694   Durbin-Watson:                   0.037
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            67965.448
Skew:                           2.104   Prob(JB):                         0.00
Kurtosis:                      18.138   Cond. No.                         3.85
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_2y   R-squared:                       0.847
Model:                                OLS   Adj. R-squared:                  0.847
Method:                     Least Squares   F-statistic:                 3.529e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:20   Log-Likelihood:                -4464.1
No. Observations:                    6356   AIC:                             8932.
Df Residuals:                        6354   BIC:                             8946.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                     0.0068      0.007      0.923      0.356      -0.008       0.021
QQQ_Rolling_Return_2y     3.1236      0.017    187.862      0.000       3.091       3.156
==============================================================================
Omnibus:                     1623.545   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4249.364
Skew:                           1.374   Prob(JB):                         0.00
Kurtosis:                       5.914   Cond. No.                         2.90
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_3y   R-squared:                       0.808
Model:                                OLS   Adj. R-squared:                  0.808
Method:                     Least Squares   F-statistic:                 2.565e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:24   Log-Likelihood:                -6738.8
No. Observations:                    6104   AIC:                         1.348e+04
Df Residuals:                        6102   BIC:                         1.350e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0608      0.013     -4.850      0.000      -0.085      -0.036
QQQ_Rolling_Return_3y     3.3335      0.021    160.141      0.000       3.293       3.374
==============================================================================
Omnibus:                      871.091   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1505.348
Skew:                           0.941   Prob(JB):                         0.00
Kurtosis:                       4.541   Cond. No.                         2.66
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_4y   R-squared:                       0.778
Model:                                OLS   Adj. R-squared:                  0.778
Method:                     Least Squares   F-statistic:                 2.048e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:28   Log-Likelihood:                -8910.6
No. Observations:                    5852   AIC:                         1.783e+04
Df Residuals:                        5850   BIC:                         1.784e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.2947      0.021    -13.729      0.000      -0.337      -0.253
QQQ_Rolling_Return_4y     3.9098      0.027    143.110      0.000       3.856       3.963
==============================================================================
Omnibus:                      221.269   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              114.248
Skew:                           0.155   Prob(JB):                     1.55e-25
Kurtosis:                       2.390   Cond. No.                         2.67
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_5y   R-squared:                       0.738
Model:                                OLS   Adj. R-squared:                  0.738
Method:                     Least Squares   F-statistic:                 1.579e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:32:31   Log-Likelihood:                -12238.
No. Observations:                    5600   AIC:                         2.448e+04
Df Residuals:                        5598   BIC:                         2.449e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.9068      0.045    -20.345      0.000      -0.994      -0.819
QQQ_Rolling_Return_5y     5.1912      0.041    125.644      0.000       5.110       5.272
==============================================================================
Omnibus:                      324.029   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              464.545
Skew:                           0.512   Prob(JB):                    1.33e-101
Kurtosis:                       3.971   Cond. No.                         2.74
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

You’re welcome to digest each plot, but here’s my observations on the above results:

1d: TQQQ tracks QQQ as expected (it’s a 3x daily return leveraged ETF after all), with a regression coefficient of 2.96 and an R^2 of 0.997, and we extrapolated half the data with the same coefficient.
1w: Essentially the same as above. A few outliers, but the regression coefficient is still 2.95 with an R^2 of 0.994. We see a slight skew toward the positive in the rolling returns.
1m: The skew toward the positive is more pronounced, and we see more outliers. The regression coefficient has decreased to 2.93 and the R^2 has dropped to 0.98, which is still very high, but we are starting to see some dispersion in the returns.
3m: The skew toward the positive is even more pronounced, and we see even more outliers. The regression coefficient has increased, to 2.98 and the R^2 has dropped to 0.96.
6m: The skew toward the positive is very pronounced, and we see a significant number of outliers with pronounced curvanture in the plot. The regression coefficient has increased again, to 3.4 and the R^2 has dropped to 0.92.
1y: At this point, based on the plot and the regression results, we can start to see that the returns of TQQQ are no longer tracking 3x the returns of QQQ as closely as they did in the shorter time periods. The regression coefficient has is now 2.84 and the R^2 has dropped to 0.88.
4y and 5y: We can see that there are periods where the rolling returns of TQQQ are significantly higher and lower than 3x the returns of QQQ, which is consistent with the idea of volatility decay.

For 4y, based on the regression results, we see that if the rolling return of QQQ was 0, then we would expect a return of -0.30 for TQQQ.

$$ r_{TQQQ} = -0.30 + 3.93 \times r_{QQQ} = -0.30 + 3.93 \times 0 = -0.30 $$

On the other end of the spectrum, if the rolling return of QQQ was 1, then we would expect a return of:

$$ r_{TQQQ} = -0.30 + 3.93 \times r_{QQQ} = -0.30 + 3.93 \times 1 = 3.63 $$

In general, the positive skew of the rolling returns of TQQQ relative to QQQ is related to the general postive return performance of QQQ. With sustained positive returns, the leverage effect of TQQQ will amplify those returns, leading to a positive skew. However, during periods of negative returns for QQQ, the leverage effect will also amplify those losses, leading to a negative skew, and to the limit of a cumulative return of -1, or a 100% loss. The overall skewness of the rolling returns will depend on the balance of these positive and negative periods.

Rolling Returns Deviation (QQQ & TQQQ) #

Next, we will the rolling returns deviation from the expected 3x return for each time period. This will give us a better picture of the volatility decay effect and how it changes over different time horizons.

rolling_returns_stats["Return_Deviation_From_3x"] = rolling_returns_stats["Slope"] - 3.0
pandas_set_decimal_places(3)
display(rolling_returns_stats.set_index("Period"))

	Intercept	Slope	R_Squared	Return Skew	Average Upside Beta	Average Downside Beta	Asymmetry	Return_Deviation_From_3x
Period
1d	-0.000	2.955	0.999	NaN	2.958	NaN	NaN	-0.045
1w	-0.001	2.953	0.994	NaN	2.557	NaN	NaN	-0.047
1m	-0.004	2.936	0.982	NaN	2.213	NaN	NaN	-0.064
3m	-0.008	2.987	0.959	NaN	1.997	-inf	inf	-0.013
6m	-0.010	3.040	0.916	-8.109	1.482	5.521	-4.039	0.040
1y	0.019	2.842	0.881	NaN	1.244	-inf	inf	-0.158
2y	0.007	3.124	0.847	36.342	1.402	12.342	-10.939	0.124
3y	-0.061	3.334	0.808	NaN	-0.049	-inf	inf	0.334
4y	-0.295	3.910	0.778	19.663	1.763	7.212	-5.449	0.910
5y	-0.907	5.191	0.738	43.272	2.421	11.480	-9.060	2.191

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Return_Deviation_From_3x"],
    title="TQQQ Deviation from Perfect 3x Leverage by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Deviation from 3x Leverage",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Slope"],
    title="TQQQ Slope by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Slope",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Intercept"],
    title="Intercept by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Intercept",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

display(rolling_returns_stats.set_index("Period"))

	Intercept	Slope	R_Squared	Return Skew	Average Upside Beta	Average Downside Beta	Asymmetry	Return_Deviation_From_3x
Period
1d	-0.000	2.955	0.999	NaN	2.958	NaN	NaN	-0.045
1w	-0.001	2.953	0.994	NaN	2.557	NaN	NaN	-0.047
1m	-0.004	2.936	0.982	NaN	2.213	NaN	NaN	-0.064
3m	-0.008	2.987	0.959	NaN	1.997	-inf	inf	-0.013
6m	-0.010	3.040	0.916	-8.109	1.482	5.521	-4.039	0.040
1y	0.019	2.842	0.881	NaN	1.244	-inf	inf	-0.158
2y	0.007	3.124	0.847	36.342	1.402	12.342	-10.939	0.124
3y	-0.061	3.334	0.808	NaN	-0.049	-inf	inf	0.334
4y	-0.295	3.910	0.778	19.663	1.763	7.212	-5.449	0.910
5y	-0.907	5.191	0.738	43.272	2.421	11.480	-9.060	2.191

This is very interesting. Up to 1 year, there is minimal difference between the mean TQQQ 1 year rolling return and the hypothetical 3x leverage, with an R^2 of greater than 0.9.

However, as we extend the time period, we see that

The “leverage factor” increases significantly, resulting in a deviation from the perfect 3x leverage.
The intercept also begins to deviate significantly from 0.

The above highlight the impact of volatility magnification over longer time horizons. This phenomenon is happening likely due to the positive returns that QQQ has achieved since 2010 - resulting in TQQQ compounding at a much higher rate than 3x - but it may and likely is not exhibited by other 3x leveraged ETFs that have not had the same positive return profile as QQQ.

With the above results, the next logical question is, when is the opportune time to buy a 3x leveraged ETF like TQQQ? To answer this, we will look a the drawdown levels of TQQQ and the subsequent returns over various time horizons.

Rolling Returns Following Drawdowns (QQQ & TQQQ) #

We will identify the drawdown levels of TQQQ and then look at the subsequent rolling returns over various time horizons.

# Copy DataFrame
qqq_tqqq_extrap_future = qqq_tqqq_extrap.copy()

# Create a list of drawdown levels to analyze
drawdown_levels = [-0.10, -0.20, -0.30, -0.40, -0.50, -0.60, -0.70, -0.80, -0.90]

# Shift the rolling return columns by the number of days in the rolling window to get the returns following the drawdown
for etf in etfs:
    for period_name, window in rolling_windows.items():
        qqq_tqqq_extrap_future[f"{etf}_Rolling_Future_Return_{period_name}"] = qqq_tqqq_extrap_future[f"{etf}_Rolling_Return_{period_name}"].shift(-window)

Now, we can analyze the future rolling returns following specific drawdown levels:

# Create a dataframe to hold rolling returns stats
rolling_returns_drawdown_stats = pd.DataFrame()

for drawdown in drawdown_levels:

    for period_name, window in rolling_windows.items():

        try:
            plot_histogram(
                df=qqq_tqqq_extrap_future[qqq_tqqq_extrap_future["TQQQ_Drawdown"] <= drawdown],
                plot_columns=[f"QQQ_Rolling_Future_Return_{period_name}", f"TQQQ_Rolling_Future_Return_{period_name}"],
                title=f"QQQ & TQQQ {period_name} Rolling Future Returns Post {drawdown} TQQQ Drawdown",
                x_label="Rolling Return",
                x_tick_spacing="Auto",
                x_tick_rotation=30,
                y_label="# Of Datapoints",
                y_tick_spacing="Auto",
                y_tick_rotation=0,
                grid=True,
                legend=True,
                export_plot=False,
                plot_file_name=None,
            )

            plot_scatter(
                df=qqq_tqqq_extrap_future[qqq_tqqq_extrap_future["TQQQ_Drawdown"] <= drawdown],
                x_plot_column=f"QQQ_Rolling_Future_Return_{period_name}",
                y_plot_columns=[f"TQQQ_Rolling_Future_Return_{period_name}"],
                title=f"QQQ & TQQQ {period_name} Rolling Future Returns Post {drawdown} TQQQ Drawdown",
                x_label="QQQ Rolling Return",
                x_format="Decimal",
                x_format_decimal_places=2,
                x_tick_spacing="Auto",
                x_tick_start=None,
                x_tick_rotation=30,
                y_label="TQQQ Rolling Return",
                y_format="Decimal",
                y_format_decimal_places=2,
                y_tick_spacing="Auto",
                y_tick_rotation=0,
                plot_OLS_regression_line=True,
                OLS_column=f"TQQQ_Rolling_Future_Return_{period_name}",
                plot_Ridge_regression_line=False,
                Ridge_column=None,
                plot_RidgeCV_regression_line=True,
                RidgeCV_column=f"TQQQ_Rolling_Future_Return_{period_name}",
                regression_constant=True,
                grid=True,
                legend=True,
                export_plot=False,
                plot_file_name=None,
            )

            # Run OLS regression with statsmodels
            model = run_regression(
                df=qqq_tqqq_extrap_future[qqq_tqqq_extrap_future["TQQQ_Drawdown"] <= drawdown],
                x_plot_column=f"QQQ_Rolling_Future_Return_{period_name}",
                y_plot_column=f"TQQQ_Rolling_Future_Return_{period_name}",
                regression_model="OLS-statsmodels",
                regression_constant=True,
            )
            print(model.summary())

            # Filter by drawdown
            drawdown_filter = qqq_tqqq_extrap_future[qqq_tqqq_extrap_future["TQQQ_Drawdown"] <= drawdown]

            # Filter by period, drop rows with missing values
            future_filter = drawdown_filter[[f"TQQQ_Rolling_Future_Return_{period_name}"]].dropna()

            # Find length of future dataframe
            future_length = len(future_filter)

            # Find length of future dataframe where return is positive
            positive_future_length = len(future_filter[future_filter[f"TQQQ_Rolling_Future_Return_{period_name}"] > 0])

            # Calculate percentage of future returns that are positive
            positive_future_percentage = (positive_future_length / future_length) if future_length > 0 else 0

            # Add the regression results to the rolling returns stats dataframe
            intercept = model.params.iloc[0]
            # intercept_pvalue = model.pvalues.iloc[0]   # p-value for Intercept
            slope = model.params.iloc[1]
            # slope_pvalue = model.pvalues.iloc[1]       # p-value for Slope
            r_squared = model.rsquared

            rolling_returns_slope_int = pd.DataFrame({
                "Drawdown": drawdown,
                "Period": period_name,
                "Intercept": [intercept],
                # "Intercept_PValue": [intercept_pvalue],
                "Slope": [slope],
                # "Slope_PValue": [slope_pvalue],
                "R_Squared": [r_squared],
                "Positive_Future_Percentage": [positive_future_percentage],
            })
            
            rolling_returns_drawdown_stats = pd.concat([rolling_returns_drawdown_stats, rolling_returns_slope_int])

        except:
            print(f"Not enough data points for drawdown level {drawdown} and period {period_name} to run regression.")

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.911e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:36   Log-Likelihood:                 33887.
No. Observations:                           6713   AIC:                        -6.777e+04
Df Residuals:                               6711   BIC:                        -6.776e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.384e-05    1.9e-05     -2.837      0.005    -9.1e-05   -1.66e-05
QQQ_Rolling_Future_Return_1d     2.9553      0.001   2628.911      0.000       2.953       2.957
==============================================================================
Omnibus:                    10002.255   Durbin-Watson:                   2.566
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         41625835.132
Skew:                          -8.172   Prob(JB):                         0.00
Kurtosis:                     388.424   Cond. No.                         59.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.107e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:39   Log-Likelihood:                 22953.
No. Observations:                           6709   AIC:                        -4.590e+04
Df Residuals:                               6707   BIC:                        -4.589e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.68e-05     -8.295      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9534      0.003   1052.034      0.000       2.948       2.959
==============================================================================
Omnibus:                     2724.426   Durbin-Watson:                   0.939
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           599408.005
Skew:                          -0.772   Prob(JB):                         0.00
Kurtosis:                      49.280   Cond. No.                         29.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.982
Model:                                       OLS   Adj. R-squared:                  0.982
Method:                            Least Squares   F-statistic:                 3.756e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:43   Log-Likelihood:                 14814.
No. Observations:                           6693   AIC:                        -2.962e+04
Df Residuals:                               6691   BIC:                        -2.961e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.258      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9366      0.005    612.886      0.000       2.927       2.946
==============================================================================
Omnibus:                     1708.914   Durbin-Watson:                   0.308
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            79259.198
Skew:                           0.422   Prob(JB):                         0.00
Kurtosis:                      19.837   Cond. No.                         14.8
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.957
Model:                                       OLS   Adj. R-squared:                  0.957
Method:                            Least Squares   F-statistic:                 1.486e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:47   Log-Likelihood:                 8047.1
No. Observations:                           6651   AIC:                        -1.609e+04
Df Residuals:                               6649   BIC:                        -1.608e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0076      0.001     -8.388      0.000      -0.009      -0.006
QQQ_Rolling_Future_Return_3m     2.9610      0.008    385.431      0.000       2.946       2.976
==============================================================================
Omnibus:                     3434.679   Durbin-Watson:                   0.113
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            87217.860
Skew:                           1.941   Prob(JB):                         0.00
Kurtosis:                      20.310   Cond. No.                         8.69
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.921
Model:                                       OLS   Adj. R-squared:                  0.921
Method:                            Least Squares   F-statistic:                 7.644e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:50   Log-Likelihood:                 3210.4
No. Observations:                           6588   AIC:                            -6417.
Df Residuals:                               6586   BIC:                            -6403.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0048      0.002     -2.481      0.013      -0.009      -0.001
QQQ_Rolling_Future_Return_6m     2.9627      0.011    276.474      0.000       2.942       2.984
==============================================================================
Omnibus:                     4217.455   Durbin-Watson:                   0.065
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           103404.384
Skew:                           2.663   Prob(JB):                         0.00
Kurtosis:                      21.664   Cond. No.                         5.87
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.894
Model:                                       OLS   Adj. R-squared:                  0.894
Method:                            Least Squares   F-statistic:                 5.421e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:54   Log-Likelihood:                -123.96
No. Observations:                           6462   AIC:                             251.9
Df Residuals:                               6460   BIC:                             265.5
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0229      0.003      6.693      0.000       0.016       0.030
QQQ_Rolling_Future_Return_1y     2.8138      0.012    232.837      0.000       2.790       2.838
==============================================================================
Omnibus:                     2629.651   Durbin-Watson:                   0.052
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            28944.026
Skew:                           1.637   Prob(JB):                         0.00
Kurtosis:                      12.837   Cond. No.                         4.00
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.847
Model:                                       OLS   Adj. R-squared:                  0.847
Method:                            Least Squares   F-statistic:                 3.430e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:32:57   Log-Likelihood:                -4303.6
No. Observations:                           6210   AIC:                             8611.
Df Residuals:                               6208   BIC:                             8625.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0215      0.008     -2.822      0.005      -0.036      -0.007
QQQ_Rolling_Future_Return_2y     3.1933      0.017    185.210      0.000       3.159       3.227
==============================================================================
Omnibus:                     1695.855   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4735.104
Skew:                           1.441   Prob(JB):                         0.00
Kurtosis:                       6.161   Cond. No.                         3.03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.814
Model:                                       OLS   Adj. R-squared:                  0.814
Method:                            Least Squares   F-statistic:                 2.606e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:01   Log-Likelihood:                -6413.7
No. Observations:                           5958   AIC:                         1.283e+04
Df Residuals:                               5956   BIC:                         1.284e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1595      0.013    -12.233      0.000      -0.185      -0.134
QQQ_Rolling_Future_Return_3y     3.5008      0.022    161.442      0.000       3.458       3.543
==============================================================================
Omnibus:                      890.338   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1586.185
Skew:                           0.965   Prob(JB):                         0.00
Kurtosis:                       4.632   Cond. No.                         2.86
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.779
Model:                                       OLS   Adj. R-squared:                  0.779
Method:                            Least Squares   F-statistic:                 2.015e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:06   Log-Likelihood:                -8622.3
No. Observations:                           5706   AIC:                         1.725e+04
Df Residuals:                               5704   BIC:                         1.726e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4278      0.023    -18.865      0.000      -0.472      -0.383
QQQ_Rolling_Future_Return_4y     4.0704      0.029    141.956      0.000       4.014       4.127
==============================================================================
Omnibus:                      148.305   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               90.147
Skew:                           0.162   Prob(JB):                     2.66e-20
Kurtosis:                       2.476   Cond. No.                         2.86
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.741
Model:                                       OLS   Adj. R-squared:                  0.741
Method:                            Least Squares   F-statistic:                 1.558e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:11   Log-Likelihood:                -11885.
No. Observations:                           5454   AIC:                         2.377e+04
Df Residuals:                               5452   BIC:                         2.379e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.1362      0.047    -24.162      0.000      -1.228      -1.044
QQQ_Rolling_Future_Return_5y     5.3839      0.043    124.822      0.000       5.299       5.469
==============================================================================
Omnibus:                      283.895   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              407.097
Skew:                           0.473   Prob(JB):                     3.98e-89
Kurtosis:                       3.946   Cond. No.                         2.92
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.673e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:15   Log-Likelihood:                 33428.
No. Observations:                           6630   AIC:                        -6.685e+04
Df Residuals:                               6628   BIC:                        -6.684e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.435e-05   1.92e-05     -2.829      0.005    -9.2e-05   -1.67e-05
QQQ_Rolling_Future_Return_1d     2.9552      0.001   2583.241      0.000       2.953       2.957
==============================================================================
Omnibus:                     9847.883   Durbin-Watson:                   2.566
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         40121425.142
Skew:                          -8.123   Prob(JB):                         0.00
Kurtosis:                     383.752   Cond. No.                         59.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.085e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:19   Log-Likelihood:                 22679.
No. Observations:                           6627   AIC:                        -4.535e+04
Df Residuals:                               6625   BIC:                        -4.534e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.73e-05     -8.044      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9524      0.003   1041.850      0.000       2.947       2.958
==============================================================================
Omnibus:                     2728.411   Durbin-Watson:                   0.932
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           607972.381
Skew:                          -0.798   Prob(JB):                         0.00
Kurtosis:                      49.896   Cond. No.                         29.2
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 3.754e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:22   Log-Likelihood:                 14767.
No. Observations:                           6616   AIC:                        -2.953e+04
Df Residuals:                               6614   BIC:                        -2.952e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.499      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9290      0.005    612.735      0.000       2.920       2.938
==============================================================================
Omnibus:                     1557.724   Durbin-Watson:                   0.313
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            79422.712
Skew:                           0.204   Prob(JB):                         0.00
Kurtosis:                      19.969   Cond. No.                         15.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.960
Model:                                       OLS   Adj. R-squared:                  0.960
Method:                            Least Squares   F-statistic:                 1.573e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:27   Log-Likelihood:                 8421.1
No. Observations:                           6574   AIC:                        -1.684e+04
Df Residuals:                               6572   BIC:                        -1.682e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0071      0.001     -8.281      0.000      -0.009      -0.005
QQQ_Rolling_Future_Return_3m     2.9138      0.007    396.593      0.000       2.899       2.928
==============================================================================
Omnibus:                     2170.483   Durbin-Watson:                   0.136
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            39171.367
Skew:                           1.112   Prob(JB):                         0.00
Kurtosis:                      14.750   Cond. No.                         8.87
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.926
Model:                                       OLS   Adj. R-squared:                  0.926
Method:                            Least Squares   F-statistic:                 8.187e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:30   Log-Likelihood:                 3840.6
No. Observations:                           6511   AIC:                            -7677.
Df Residuals:                               6509   BIC:                            -7664.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0021      0.002     -1.215      0.224      -0.006       0.001
QQQ_Rolling_Future_Return_6m     2.8746      0.010    286.133      0.000       2.855       2.894
==============================================================================
Omnibus:                     3235.775   Durbin-Watson:                   0.076
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            51105.740
Skew:                           1.991   Prob(JB):                         0.00
Kurtosis:                      16.135   Cond. No.                         6.06
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.899
Model:                                       OLS   Adj. R-squared:                  0.899
Method:                            Least Squares   F-statistic:                 5.654e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:34   Log-Likelihood:                 129.52
No. Observations:                           6385   AIC:                            -255.0
Df Residuals:                               6383   BIC:                            -241.5
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0227      0.003      6.858      0.000       0.016       0.029
QQQ_Rolling_Future_Return_1y     2.8392      0.012    237.784      0.000       2.816       2.863
==============================================================================
Omnibus:                     2422.758   Durbin-Watson:                   0.068
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            19269.888
Skew:                           1.604   Prob(JB):                         0.00
Kurtosis:                      10.883   Cond. No.                         4.09
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.846
Model:                                       OLS   Adj. R-squared:                  0.846
Method:                            Least Squares   F-statistic:                 3.369e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:39   Log-Likelihood:                -4224.9
No. Observations:                           6133   AIC:                             8454.
Df Residuals:                               6131   BIC:                             8467.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0315      0.008     -4.051      0.000      -0.047      -0.016
QQQ_Rolling_Future_Return_2y     3.2194      0.018    183.536      0.000       3.185       3.254
==============================================================================
Omnibus:                     1720.128   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4955.786
Skew:                           1.467   Prob(JB):                         0.00
Kurtosis:                       6.284   Cond. No.                         3.09
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.817
Model:                                       OLS   Adj. R-squared:                  0.817
Method:                            Least Squares   F-statistic:                 2.632e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:42   Log-Likelihood:                -6236.8
No. Observations:                           5881   AIC:                         1.248e+04
Df Residuals:                               5879   BIC:                         1.249e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2171      0.013    -16.283      0.000      -0.243      -0.191
QQQ_Rolling_Future_Return_3y     3.5972      0.022    162.238      0.000       3.554       3.641
==============================================================================
Omnibus:                      898.619   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1636.684
Skew:                           0.974   Prob(JB):                         0.00
Kurtosis:                       4.699   Cond. No.                         2.98
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.780
Model:                                       OLS   Adj. R-squared:                  0.780
Method:                            Least Squares   F-statistic:                 1.998e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:46   Log-Likelihood:                -8466.9
No. Observations:                           5629   AIC:                         1.694e+04
Df Residuals:                               5627   BIC:                         1.695e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5060      0.023    -21.632      0.000      -0.552      -0.460
QQQ_Rolling_Future_Return_4y     4.1639      0.029    141.365      0.000       4.106       4.222
==============================================================================
Omnibus:                      110.238   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               75.087
Skew:                           0.165   Prob(JB):                     4.95e-17
Kurtosis:                       2.541   Cond. No.                         2.97
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.742
Model:                                       OLS   Adj. R-squared:                  0.742
Method:                            Least Squares   F-statistic:                 1.547e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:51   Log-Likelihood:                -11697.
No. Observations:                           5377   AIC:                         2.340e+04
Df Residuals:                               5375   BIC:                         2.341e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.2669      0.048    -26.156      0.000      -1.362      -1.172
QQQ_Rolling_Future_Return_5y     5.4927      0.044    124.366      0.000       5.406       5.579
==============================================================================
Omnibus:                      263.633   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              380.037
Skew:                           0.451   Prob(JB):                     2.99e-83
Kurtosis:                       3.939   Cond. No.                         3.02
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.487e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:55   Log-Likelihood:                 33121.
No. Observations:                           6574   AIC:                        -6.624e+04
Df Residuals:                               6572   BIC:                        -6.622e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.443e-05   1.94e-05     -2.811      0.005   -9.24e-05   -1.65e-05
QQQ_Rolling_Future_Return_1d     2.9553      0.001   2546.890      0.000       2.953       2.958
==============================================================================
Omnibus:                     9750.662   Durbin-Watson:                   2.564
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         39259436.337
Skew:                          -8.100   Prob(JB):                         0.00
Kurtosis:                     381.238   Cond. No.                         59.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.133e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:33:59   Log-Likelihood:                 22693.
No. Observations:                           6573   AIC:                        -4.538e+04
Df Residuals:                               6571   BIC:                        -4.537e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.47e-05     -8.330      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9560      0.003   1064.455      0.000       2.951       2.961
==============================================================================
Omnibus:                     3486.989   Durbin-Watson:                   0.902
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           393847.121
Skew:                          -1.577   Prob(JB):                         0.00
Kurtosis:                      40.790   Cond. No.                         29.4
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 3.724e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:02   Log-Likelihood:                 14721.
No. Observations:                           6570   AIC:                        -2.944e+04
Df Residuals:                               6568   BIC:                        -2.943e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.644      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9245      0.005    610.236      0.000       2.915       2.934
==============================================================================
Omnibus:                     1526.763   Durbin-Watson:                   0.304
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            80156.566
Skew:                           0.132   Prob(JB):                         0.00
Kurtosis:                      20.110   Cond. No.                         15.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.614e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:06   Log-Likelihood:                 8557.4
No. Observations:                           6534   AIC:                        -1.711e+04
Df Residuals:                               6532   BIC:                        -1.710e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0068      0.001     -8.136      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8980      0.007    401.740      0.000       2.884       2.912
==============================================================================
Omnibus:                     1370.505   Durbin-Watson:                   0.103
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15604.976
Skew:                           0.673   Prob(JB):                         0.00
Kurtosis:                      10.450   Cond. No.                         8.93
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.929
Model:                                       OLS   Adj. R-squared:                  0.929
Method:                            Least Squares   F-statistic:                 8.467e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:10   Log-Likelihood:                 4192.8
No. Observations:                           6471   AIC:                            -8382.
Df Residuals:                               6469   BIC:                            -8368.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0011      0.002     -0.693      0.488      -0.004       0.002
QQQ_Rolling_Future_Return_6m     2.8227      0.010    290.977      0.000       2.804       2.842
==============================================================================
Omnibus:                     2729.155   Durbin-Watson:                   0.109
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            37443.267
Skew:                           1.642   Prob(JB):                         0.00
Kurtosis:                      14.318   Cond. No.                         6.18
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.902
Model:                                       OLS   Adj. R-squared:                  0.902
Method:                            Least Squares   F-statistic:                 5.862e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:14   Log-Likelihood:                 306.92
No. Observations:                           6345   AIC:                            -609.8
Df Residuals:                               6343   BIC:                            -596.3
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0223      0.003      6.928      0.000       0.016       0.029
QQQ_Rolling_Future_Return_1y     2.8563      0.012    242.124      0.000       2.833       2.879
==============================================================================
Omnibus:                     1986.105   Durbin-Watson:                   0.043
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8512.500
Skew:                           1.479   Prob(JB):                         0.00
Kurtosis:                       7.842   Cond. No.                         4.14
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.846
Model:                                       OLS   Adj. R-squared:                  0.846
Method:                            Least Squares   F-statistic:                 3.335e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:18   Log-Likelihood:                -4185.3
No. Observations:                           6093   AIC:                             8375.
Df Residuals:                               6091   BIC:                             8388.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0344      0.008     -4.390      0.000      -0.050      -0.019
QQQ_Rolling_Future_Return_2y     3.2283      0.018    182.610      0.000       3.194       3.263
==============================================================================
Omnibus:                     1727.215   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5044.840
Skew:                           1.477   Prob(JB):                         0.00
Kurtosis:                       6.338   Cond. No.                         3.11
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.820
Model:                                       OLS   Adj. R-squared:                  0.819
Method:                            Least Squares   F-statistic:                 2.651e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:22   Log-Likelihood:                -6139.3
No. Observations:                           5841   AIC:                         1.228e+04
Df Residuals:                               5839   BIC:                         1.230e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2500      0.013    -18.534      0.000      -0.276      -0.224
QQQ_Rolling_Future_Return_3y     3.6520      0.022    162.821      0.000       3.608       3.696
==============================================================================
Omnibus:                      902.038   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1667.762
Skew:                           0.976   Prob(JB):                         0.00
Kurtosis:                       4.744   Cond. No.                         3.04
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.781
Model:                                       OLS   Adj. R-squared:                  0.781
Method:                            Least Squares   F-statistic:                 1.990e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:25   Log-Likelihood:                -8384.8
No. Observations:                           5589   AIC:                         1.677e+04
Df Residuals:                               5587   BIC:                         1.679e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5491      0.024    -23.091      0.000      -0.596      -0.503
QQQ_Rolling_Future_Return_4y     4.2152      0.030    141.074      0.000       4.157       4.274
==============================================================================
Omnibus:                       91.895   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               66.824
Skew:                           0.167   Prob(JB):                     3.09e-15
Kurtosis:                       2.581   Cond. No.                         3.03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.743
Model:                                       OLS   Adj. R-squared:                  0.743
Method:                            Least Squares   F-statistic:                 1.541e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:29   Log-Likelihood:                -11599.
No. Observations:                           5337   AIC:                         2.320e+04
Df Residuals:                               5335   BIC:                         2.321e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3392      0.049    -27.216      0.000      -1.436      -1.243
QQQ_Rolling_Future_Return_5y     5.5527      0.045    124.156      0.000       5.465       5.640
==============================================================================
Omnibus:                      252.830   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              366.376
Skew:                           0.438   Prob(JB):                     2.77e-80
Kurtosis:                       3.938   Cond. No.                         3.07
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.423e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:33   Log-Likelihood:                 33007.
No. Observations:                           6553   AIC:                        -6.601e+04
Df Residuals:                               6551   BIC:                        -6.600e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         -5.37e-05   1.94e-05     -2.765      0.006   -9.18e-05   -1.56e-05
QQQ_Rolling_Future_Return_1d     2.9553      0.001   2534.297      0.000       2.953       2.958
==============================================================================
Omnibus:                     9718.348   Durbin-Watson:                   2.566
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         39023258.345
Skew:                          -8.099   Prob(JB):                         0.00
Kurtosis:                     380.701   Cond. No.                         60.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.129e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:38   Log-Likelihood:                 22667.
No. Observations:                           6553   AIC:                        -4.533e+04
Df Residuals:                               6551   BIC:                        -4.532e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.42e-05     -8.569      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9541      0.003   1062.729      0.000       2.949       2.960
==============================================================================
Omnibus:                     3613.538   Durbin-Watson:                   0.881
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           408611.855
Skew:                          -1.685   Prob(JB):                         0.00
Kurtosis:                      41.538   Cond. No.                         29.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 3.704e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:43   Log-Likelihood:                 14702.
No. Observations:                           6553   AIC:                        -2.940e+04
Df Residuals:                               6551   BIC:                        -2.939e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.643      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9216      0.005    608.577      0.000       2.912       2.931
==============================================================================
Omnibus:                     1521.866   Durbin-Watson:                   0.293
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            80945.795
Skew:                           0.112   Prob(JB):                         0.00
Kurtosis:                      20.217   Cond. No.                         15.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.620e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:47   Log-Likelihood:                 8571.4
No. Observations:                           6523   AIC:                        -1.714e+04
Df Residuals:                               6521   BIC:                        -1.713e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0066      0.001     -7.965      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8962      0.007    402.551      0.000       2.882       2.910
==============================================================================
Omnibus:                     1401.017   Durbin-Watson:                   0.101
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15929.868
Skew:                           0.699   Prob(JB):                         0.00
Kurtosis:                      10.527   Cond. No.                         8.94
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.931
Model:                                       OLS   Adj. R-squared:                  0.931
Method:                            Least Squares   F-statistic:                 8.747e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:51   Log-Likelihood:                 4370.9
No. Observations:                           6460   AIC:                            -8738.
Df Residuals:                               6458   BIC:                            -8724.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.002     -0.363      0.717      -0.004       0.003
QQQ_Rolling_Future_Return_6m     2.8041      0.009    295.754      0.000       2.786       2.823
==============================================================================
Omnibus:                     1674.962   Durbin-Watson:                   0.057
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8401.998
Skew:                           1.159   Prob(JB):                         0.00
Kurtosis:                       8.084   Cond. No.                         6.21
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.903
Model:                                       OLS   Adj. R-squared:                  0.903
Method:                            Least Squares   F-statistic:                 5.905e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:55   Log-Likelihood:                 341.65
No. Observations:                           6334   AIC:                            -679.3
Df Residuals:                               6332   BIC:                            -665.8
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0210      0.003      6.538      0.000       0.015       0.027
QQQ_Rolling_Future_Return_1y     2.8665      0.012    242.992      0.000       2.843       2.890
==============================================================================
Omnibus:                     1983.991   Durbin-Watson:                   0.040
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8504.663
Skew:                           1.480   Prob(JB):                         0.00
Kurtosis:                       7.844   Cond. No.                         4.16
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.846
Model:                                       OLS   Adj. R-squared:                  0.846
Method:                            Least Squares   F-statistic:                 3.336e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:34:59   Log-Likelihood:                -4165.6
No. Observations:                           6082   AIC:                             8335.
Df Residuals:                               6080   BIC:                             8349.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0372      0.008     -4.736      0.000      -0.053      -0.022
QQQ_Rolling_Future_Return_2y     3.2356      0.018    182.653      0.000       3.201       3.270
==============================================================================
Omnibus:                     1735.957   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5121.189
Skew:                           1.483   Prob(JB):                         0.00
Kurtosis:                       6.378   Cond. No.                         3.12
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.821
Model:                                       OLS   Adj. R-squared:                  0.821
Method:                            Least Squares   F-statistic:                 2.666e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:04   Log-Likelihood:                -6103.9
No. Observations:                           5830   AIC:                         1.221e+04
Df Residuals:                               5828   BIC:                         1.223e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2615      0.014    -19.338      0.000      -0.288      -0.235
QQQ_Rolling_Future_Return_3y     3.6714      0.022    163.272      0.000       3.627       3.715
==============================================================================
Omnibus:                      895.755   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1658.417
Skew:                           0.971   Prob(JB):                         0.00
Kurtosis:                       4.748   Cond. No.                         3.06
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.781
Model:                                       OLS   Adj. R-squared:                  0.781
Method:                            Least Squares   F-statistic:                 1.992e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:08   Log-Likelihood:                -8358.0
No. Observations:                           5578   AIC:                         1.672e+04
Df Residuals:                               5576   BIC:                         1.673e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5638      0.024    -23.606      0.000      -0.611      -0.517
QQQ_Rolling_Future_Return_4y     4.2329      0.030    141.127      0.000       4.174       4.292
==============================================================================
Omnibus:                       86.160   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               63.664
Skew:                           0.165   Prob(JB):                     1.50e-14
Kurtosis:                       2.593   Cond. No.                         3.05
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.743
Model:                                       OLS   Adj. R-squared:                  0.743
Method:                            Least Squares   F-statistic:                 1.543e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:12   Log-Likelihood:                -11568.
No. Observations:                           5326   AIC:                         2.314e+04
Df Residuals:                               5324   BIC:                         2.315e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3642      0.049    -27.608      0.000      -1.461      -1.267
QQQ_Rolling_Future_Return_5y     5.5737      0.045    124.205      0.000       5.486       5.662
==============================================================================
Omnibus:                      248.125   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              360.908
Skew:                           0.432   Prob(JB):                     4.26e-79
Kurtosis:                       3.938   Cond. No.                         3.09
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.278e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:16   Log-Likelihood:                 32614.
No. Observations:                           6480   AIC:                        -6.522e+04
Df Residuals:                               6478   BIC:                        -6.521e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         -4.98e-05   1.96e-05     -2.540      0.011   -8.82e-05   -1.14e-05
QQQ_Rolling_Future_Return_1d     2.9551      0.001   2505.632      0.000       2.953       2.957
==============================================================================
Omnibus:                     9606.715   Durbin-Watson:                   2.568
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         38196819.425
Skew:                          -8.095   Prob(JB):                         0.00
Kurtosis:                     378.776   Cond. No.                         60.2
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.118e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:20   Log-Likelihood:                 22412.
No. Observations:                           6480   AIC:                        -4.482e+04
Df Residuals:                               6478   BIC:                        -4.481e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.48e-05     -8.165      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9535      0.003   1057.400      0.000       2.948       2.959
==============================================================================
Omnibus:                     3552.509   Durbin-Watson:                   0.887
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           405301.538
Skew:                          -1.666   Prob(JB):                         0.00
Kurtosis:                      41.601   Cond. No.                         29.5
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 3.642e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:24   Log-Likelihood:                 14518.
No. Observations:                           6480   AIC:                        -2.903e+04
Df Residuals:                               6478   BIC:                        -2.902e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0033      0.000    -10.350      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9199      0.005    603.449      0.000       2.910       2.929
==============================================================================
Omnibus:                     1498.953   Durbin-Watson:                   0.292
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            79274.697
Skew:                           0.099   Prob(JB):                         0.00
Kurtosis:                      20.134   Cond. No.                         15.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.611e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:28   Log-Likelihood:                 8483.0
No. Observations:                           6462   AIC:                        -1.696e+04
Df Residuals:                               6460   BIC:                        -1.695e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0062      0.001     -7.423      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8955      0.007    401.360      0.000       2.881       2.910
==============================================================================
Omnibus:                     1385.700   Durbin-Watson:                   0.101
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15646.097
Skew:                           0.699   Prob(JB):                         0.00
Kurtosis:                      10.494   Cond. No.                         8.91
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.931
Model:                                       OLS   Adj. R-squared:                  0.931
Method:                            Least Squares   F-statistic:                 8.720e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:32   Log-Likelihood:                 4343.3
No. Observations:                           6424   AIC:                            -8683.
Df Residuals:                               6422   BIC:                            -8669.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002      0.002     -0.130      0.897      -0.003       0.003
QQQ_Rolling_Future_Return_6m     2.8045      0.009    295.300      0.000       2.786       2.823
==============================================================================
Omnibus:                     1672.145   Durbin-Watson:                   0.055
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8272.538
Skew:                           1.168   Prob(JB):                         0.00
Kurtosis:                       8.045   Cond. No.                         6.20
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.903
Model:                                       OLS   Adj. R-squared:                  0.903
Method:                            Least Squares   F-statistic:                 5.911e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:38   Log-Likelihood:                 349.34
No. Observations:                           6331   AIC:                            -694.7
Df Residuals:                               6329   BIC:                            -681.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0205      0.003      6.376      0.000       0.014       0.027
QQQ_Rolling_Future_Return_1y     2.8691      0.012    243.131      0.000       2.846       2.892
==============================================================================
Omnibus:                     1982.778   Durbin-Watson:                   0.038
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8522.330
Skew:                           1.479   Prob(JB):                         0.00
Kurtosis:                       7.854   Cond. No.                         4.16
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.846
Model:                                       OLS   Adj. R-squared:                  0.846
Method:                            Least Squares   F-statistic:                 3.339e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:42   Log-Likelihood:                -4158.4
No. Observations:                           6079   AIC:                             8321.
Df Residuals:                               6077   BIC:                             8334.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0387      0.008     -4.920      0.000      -0.054      -0.023
QQQ_Rolling_Future_Return_2y     3.2392      0.018    182.723      0.000       3.204       3.274
==============================================================================
Omnibus:                     1741.033   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5160.249
Skew:                           1.487   Prob(JB):                         0.00
Kurtosis:                       6.396   Cond. No.                         3.13
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.821
Model:                                       OLS   Adj. R-squared:                  0.821
Method:                            Least Squares   F-statistic:                 2.673e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:46   Log-Likelihood:                -6091.1
No. Observations:                           5827   AIC:                         1.219e+04
Df Residuals:                               5825   BIC:                         1.220e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2656      0.014    -19.632      0.000      -0.292      -0.239
QQQ_Rolling_Future_Return_3y     3.6783      0.022    163.500      0.000       3.634       3.722
==============================================================================
Omnibus:                      892.642   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1653.181
Skew:                           0.968   Prob(JB):                         0.00
Kurtosis:                       4.749   Cond. No.                         3.07
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.782
Model:                                       OLS   Adj. R-squared:                  0.781
Method:                            Least Squares   F-statistic:                 1.993e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:50   Log-Likelihood:                -8349.3
No. Observations:                           5575   AIC:                         1.670e+04
Df Residuals:                               5573   BIC:                         1.672e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5688      0.024    -23.786      0.000      -0.616      -0.522
QQQ_Rolling_Future_Return_4y     4.2389      0.030    141.186      0.000       4.180       4.298
==============================================================================
Omnibus:                       84.267   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               62.495
Skew:                           0.163   Prob(JB):                     2.69e-14
Kurtosis:                       2.597   Cond. No.                         3.06
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.744
Model:                                       OLS   Adj. R-squared:                  0.744
Method:                            Least Squares   F-statistic:                 1.544e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:55   Log-Likelihood:                -11558.
No. Observations:                           5323   AIC:                         2.312e+04
Df Residuals:                               5321   BIC:                         2.313e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3730      0.049    -27.756      0.000      -1.470      -1.276
QQQ_Rolling_Future_Return_5y     5.5813      0.045    124.262      0.000       5.493       5.669
==============================================================================
Omnibus:                      246.200   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              358.803
Skew:                           0.429   Prob(JB):                     1.22e-78
Kurtosis:                       3.939   Cond. No.                         3.09
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 6.009e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:35:59   Log-Likelihood:                 31565.
No. Observations:                           6286   AIC:                        -6.313e+04
Df Residuals:                               6284   BIC:                        -6.311e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -4.276e-05   2.01e-05     -2.123      0.034   -8.22e-05   -3.27e-06
QQQ_Rolling_Future_Return_1d     2.9548      0.001   2451.416      0.000       2.952       2.957
==============================================================================
Omnibus:                     9298.069   Durbin-Watson:                   2.569
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         35800414.567
Skew:                          -8.062   Prob(JB):                         0.00
Kurtosis:                     372.359   Cond. No.                         59.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.078e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:03   Log-Likelihood:                 21698.
No. Observations:                           6286   AIC:                        -4.339e+04
Df Residuals:                               6284   BIC:                        -4.338e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0007    9.7e-05     -7.558      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9526      0.003   1038.340      0.000       2.947       2.958
==============================================================================
Omnibus:                     3412.046   Durbin-Watson:                   0.895
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           385507.507
Skew:                          -1.639   Prob(JB):                         0.00
Kurtosis:                      41.225   Cond. No.                         29.4
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.982
Model:                                       OLS   Adj. R-squared:                  0.982
Method:                            Least Squares   F-statistic:                 3.507e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:07   Log-Likelihood:                 14066.
No. Observations:                           6286   AIC:                        -2.813e+04
Df Residuals:                               6284   BIC:                        -2.811e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0032      0.000     -9.681      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9159      0.005    592.181      0.000       2.906       2.926
==============================================================================
Omnibus:                     1452.213   Durbin-Watson:                   0.296
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            77489.348
Skew:                           0.072   Prob(JB):                         0.00
Kurtosis:                      20.200   Cond. No.                         15.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.567e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:11   Log-Likelihood:                 8267.6
No. Observations:                           6282   AIC:                        -1.653e+04
Df Residuals:                               6280   BIC:                        -1.652e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0059      0.001     -7.041      0.000      -0.008      -0.004
QQQ_Rolling_Future_Return_3m     2.8978      0.007    395.838      0.000       2.883       2.912
==============================================================================
Omnibus:                     1352.723   Durbin-Watson:                   0.102
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15617.746
Skew:                           0.696   Prob(JB):                         0.00
Kurtosis:                      10.598   Cond. No.                         8.95
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.932
Model:                                       OLS   Adj. R-squared:                  0.932
Method:                            Least Squares   F-statistic:                 8.620e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:15   Log-Likelihood:                 4283.9
No. Observations:                           6282   AIC:                            -8564.
Df Residuals:                               6280   BIC:                            -8550.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002      0.002     -0.119      0.905      -0.003       0.003
QQQ_Rolling_Future_Return_6m     2.8190      0.010    293.591      0.000       2.800       2.838
==============================================================================
Omnibus:                     1637.953   Durbin-Watson:                   0.057
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8397.666
Skew:                           1.159   Prob(JB):                         0.00
Kurtosis:                       8.168   Cond. No.                         6.24
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.905
Model:                                       OLS   Adj. R-squared:                  0.905
Method:                            Least Squares   F-statistic:                 5.967e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:19   Log-Likelihood:                 422.25
No. Observations:                           6265   AIC:                            -840.5
Df Residuals:                               6263   BIC:                            -827.0
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0161      0.003      5.020      0.000       0.010       0.022
QQQ_Rolling_Future_Return_1y     2.8965      0.012    244.270      0.000       2.873       2.920
==============================================================================
Omnibus:                     1968.116   Durbin-Watson:                   0.039
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8688.728
Skew:                           1.474   Prob(JB):                         0.00
Kurtosis:                       7.959   Cond. No.                         4.22
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.849
Model:                                       OLS   Adj. R-squared:                  0.849
Method:                            Least Squares   F-statistic:                 3.403e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:22   Log-Likelihood:                -4055.4
No. Observations:                           6043   AIC:                             8115.
Df Residuals:                               6041   BIC:                             8128.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0531      0.008     -6.749      0.000      -0.069      -0.038
QQQ_Rolling_Future_Return_2y     3.2773      0.018    184.459      0.000       3.242       3.312
==============================================================================
Omnibus:                     1751.026   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5355.377
Skew:                           1.491   Prob(JB):                         0.00
Kurtosis:                       6.519   Cond. No.                         3.17
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.824
Model:                                       OLS   Adj. R-squared:                  0.824
Method:                            Least Squares   F-statistic:                 2.717e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:26   Log-Likelihood:                -5989.9
No. Observations:                           5791   AIC:                         1.198e+04
Df Residuals:                               5789   BIC:                         1.200e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2887      0.014    -21.256      0.000      -0.315      -0.262
QQQ_Rolling_Future_Return_3y     3.7207      0.023    164.844      0.000       3.676       3.765
==============================================================================
Omnibus:                      873.964   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1632.502
Skew:                           0.952   Prob(JB):                         0.00
Kurtosis:                       4.772   Cond. No.                         3.12
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.784
Model:                                       OLS   Adj. R-squared:                  0.784
Method:                            Least Squares   F-statistic:                 2.014e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:30   Log-Likelihood:                -8253.1
No. Observations:                           5539   AIC:                         1.651e+04
Df Residuals:                               5537   BIC:                         1.652e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.6009      0.024    -24.966      0.000      -0.648      -0.554
QQQ_Rolling_Future_Return_4y     4.2830      0.030    141.933      0.000       4.224       4.342
==============================================================================
Omnibus:                       68.621   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               51.608
Skew:                           0.143   Prob(JB):                     6.22e-12
Kurtosis:                       2.624   Cond. No.                         3.10
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.746
Model:                                       OLS   Adj. R-squared:                  0.746
Method:                            Least Squares   F-statistic:                 1.555e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:34   Log-Likelihood:                -11492.
No. Observations:                           5303   AIC:                         2.299e+04
Df Residuals:                               5301   BIC:                         2.300e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.4342      0.050    -28.784      0.000      -1.532      -1.337
QQQ_Rolling_Future_Return_5y     5.6336      0.045    124.683      0.000       5.545       5.722
==============================================================================
Omnibus:                      232.608   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              344.101
Skew:                           0.408   Prob(JB):                     1.90e-75
Kurtosis:                       3.945   Cond. No.                         3.13
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 5.380e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:39   Log-Likelihood:                 29448.
No. Observations:                           5891   AIC:                        -5.889e+04
Df Residuals:                               5889   BIC:                        -5.888e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -3.067e-05   2.13e-05     -1.441      0.150   -7.24e-05    1.11e-05
QQQ_Rolling_Future_Return_1d     2.9545      0.001   2319.462      0.000       2.952       2.957
==============================================================================
Omnibus:                     8699.984   Durbin-Watson:                   2.576
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         31790095.902
Skew:                          -8.044   Prob(JB):                         0.00
Kurtosis:                     362.520   Cond. No.                         59.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 1.008e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:43   Log-Likelihood:                 20307.
No. Observations:                           5891   AIC:                        -4.061e+04
Df Residuals:                               5889   BIC:                        -4.060e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.000     -6.376      0.000      -0.001      -0.000
QQQ_Rolling_Future_Return_1w     2.9526      0.003   1003.872      0.000       2.947       2.958
==============================================================================
Omnibus:                     3288.180   Durbin-Watson:                   0.897
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           369125.880
Skew:                          -1.719   Prob(JB):                         0.00
Kurtosis:                      41.626   Cond. No.                         29.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 3.372e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:47   Log-Likelihood:                 13212.
No. Observations:                           5891   AIC:                        -2.642e+04
Df Residuals:                               5889   BIC:                        -2.641e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0025      0.000     -7.390      0.000      -0.003      -0.002
QQQ_Rolling_Future_Return_1m     2.9140      0.005    580.700      0.000       2.904       2.924
==============================================================================
Omnibus:                     1415.634   Durbin-Watson:                   0.309
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            78310.905
Skew:                           0.194   Prob(JB):                         0.00
Kurtosis:                      20.857   Cond. No.                         15.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.962
Model:                                       OLS   Adj. R-squared:                  0.962
Method:                            Least Squares   F-statistic:                 1.509e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:51   Log-Likelihood:                 7801.2
No. Observations:                           5891   AIC:                        -1.560e+04
Df Residuals:                               5889   BIC:                        -1.559e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0040      0.001     -4.630      0.000      -0.006      -0.002
QQQ_Rolling_Future_Return_3m     2.9068      0.007    388.474      0.000       2.892       2.921
==============================================================================
Omnibus:                     1368.852   Durbin-Watson:                   0.107
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            16453.777
Skew:                           0.766   Prob(JB):                         0.00
Kurtosis:                      11.043   Cond. No.                         8.93
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.934
Model:                                       OLS   Adj. R-squared:                  0.934
Method:                            Least Squares   F-statistic:                 8.379e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:55   Log-Likelihood:                 4199.5
No. Observations:                           5891   AIC:                            -8395.
Df Residuals:                               5889   BIC:                            -8382.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0018      0.002     -1.061      0.289      -0.005       0.001
QQQ_Rolling_Future_Return_6m     2.8731      0.010    289.466      0.000       2.854       2.893
==============================================================================
Omnibus:                     1462.858   Durbin-Watson:                   0.063
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8114.729
Skew:                           1.074   Prob(JB):                         0.00
Kurtosis:                       8.333   Cond. No.                         6.45
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.912
Model:                                       OLS   Adj. R-squared:                  0.912
Method:                            Least Squares   F-statistic:                 6.137e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:36:59   Log-Likelihood:                 724.28
No. Observations:                           5891   AIC:                            -1445.
Df Residuals:                               5889   BIC:                            -1431.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0012      0.003     -0.381      0.703      -0.008       0.005
QQQ_Rolling_Future_Return_1y     3.0156      0.012    247.739      0.000       2.992       3.039
==============================================================================
Omnibus:                     1762.243   Durbin-Watson:                   0.044
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8262.363
Skew:                           1.376   Prob(JB):                         0.00
Kurtosis:                       8.108   Cond. No.                         4.45
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.864
Model:                                       OLS   Adj. R-squared:                  0.864
Method:                            Least Squares   F-statistic:                 3.704e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:03   Log-Likelihood:                -3557.0
No. Observations:                           5835   AIC:                             7118.
Df Residuals:                               5833   BIC:                             7131.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1141      0.008    -14.432      0.000      -0.130      -0.099
QQQ_Rolling_Future_Return_2y     3.4455      0.018    192.447      0.000       3.410       3.481
==============================================================================
Omnibus:                     1666.636   Durbin-Watson:                   0.020
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5633.507
Skew:                           1.426   Prob(JB):                         0.00
Kurtosis:                       6.878   Cond. No.                         3.37
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.840
Model:                                       OLS   Adj. R-squared:                  0.840
Method:                            Least Squares   F-statistic:                 2.942e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:07   Log-Likelihood:                -5490.2
No. Observations:                           5597   AIC:                         1.098e+04
Df Residuals:                               5595   BIC:                         1.100e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3824      0.014    -27.778      0.000      -0.409      -0.355
QQQ_Rolling_Future_Return_3y     3.9011      0.023    171.519      0.000       3.857       3.946
==============================================================================
Omnibus:                      704.792   Durbin-Watson:                   0.016
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1332.452
Skew:                           0.810   Prob(JB):                    4.59e-290
Kurtosis:                       4.757   Cond. No.                         3.30
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.801
Model:                                       OLS   Adj. R-squared:                  0.801
Method:                            Least Squares   F-statistic:                 2.144e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:11   Log-Likelihood:                -7744.0
No. Observations:                           5345   AIC:                         1.549e+04
Df Residuals:                               5343   BIC:                         1.551e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.7291      0.025    -29.679      0.000      -0.777      -0.681
QQQ_Rolling_Future_Return_4y     4.4752      0.031    146.441      0.000       4.415       4.535
==============================================================================
Omnibus:                       21.704   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               16.422
Skew:                           0.009   Prob(JB):                     0.000272
Kurtosis:                       2.729   Cond. No.                         3.26
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.755
Model:                                       OLS   Adj. R-squared:                  0.755
Method:                            Least Squares   F-statistic:                 1.605e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:15   Log-Likelihood:                -11217.
No. Observations:                           5223   AIC:                         2.244e+04
Df Residuals:                               5221   BIC:                         2.245e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.6991      0.051    -33.137      0.000      -1.800      -1.599
QQQ_Rolling_Future_Return_5y     5.8591      0.046    126.703      0.000       5.768       5.950
==============================================================================
Omnibus:                      170.924   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              276.130
Skew:                           0.299   Prob(JB):                     1.09e-60
Kurtosis:                       3.955   Cond. No.                         3.29
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 4.767e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:20   Log-Likelihood:                 27167.
No. Observations:                           5450   AIC:                        -5.433e+04
Df Residuals:                               5448   BIC:                        -5.432e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -1.603e-05   2.24e-05     -0.715      0.475      -6e-05     2.8e-05
QQQ_Rolling_Future_Return_1d     2.9532      0.001   2183.322      0.000       2.951       2.956
==============================================================================
Omnibus:                     8184.686   Durbin-Watson:                   2.578
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         30358332.792
Skew:                          -8.327   Prob(JB):                         0.00
Kurtosis:                     368.254   Cond. No.                         60.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 9.352e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:24   Log-Likelihood:                 18855.
No. Observations:                           5450   AIC:                        -3.771e+04
Df Residuals:                               5448   BIC:                        -3.769e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.000     -5.513      0.000      -0.001      -0.000
QQQ_Rolling_Future_Return_1w     2.9496      0.003    967.067      0.000       2.944       2.956
==============================================================================
Omnibus:                     3225.323   Durbin-Watson:                   0.872
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           383461.293
Skew:                          -1.880   Prob(JB):                         0.00
Kurtosis:                      43.921   Cond. No.                         29.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.982
Model:                                       OLS   Adj. R-squared:                  0.982
Method:                            Least Squares   F-statistic:                 3.035e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:28   Log-Likelihood:                 12190.
No. Observations:                           5450   AIC:                        -2.438e+04
Df Residuals:                               5448   BIC:                        -2.436e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0021      0.000     -5.912      0.000      -0.003      -0.001
QQQ_Rolling_Future_Return_1m     2.9064      0.005    550.928      0.000       2.896       2.917
==============================================================================
Omnibus:                     1331.295   Durbin-Watson:                   0.299
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            76820.025
Skew:                           0.205   Prob(JB):                         0.00
Kurtosis:                      21.388   Cond. No.                         15.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.352e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:33   Log-Likelihood:                 7159.6
No. Observations:                           5450   AIC:                        -1.432e+04
Df Residuals:                               5448   BIC:                        -1.430e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0036      0.001     -3.911      0.000      -0.005      -0.002
QQQ_Rolling_Future_Return_3m     2.9142      0.008    367.681      0.000       2.899       2.930
==============================================================================
Omnibus:                     1265.114   Durbin-Watson:                   0.097
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15817.724
Skew:                           0.752   Prob(JB):                         0.00
Kurtosis:                      11.209   Cond. No.                         9.00
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.937
Model:                                       OLS   Adj. R-squared:                  0.937
Method:                            Least Squares   F-statistic:                 8.117e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:37   Log-Likelihood:                 4027.0
No. Observations:                           5450   AIC:                            -8050.
Df Residuals:                               5448   BIC:                            -8037.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0043      0.002     -2.531      0.011      -0.008      -0.001
QQQ_Rolling_Future_Return_6m     2.9104      0.010    284.897      0.000       2.890       2.930
==============================================================================
Omnibus:                      970.680   Durbin-Watson:                   0.061
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4547.099
Skew:                           0.789   Prob(JB):                         0.00
Kurtosis:                       7.187   Cond. No.                         6.55
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.917
Model:                                       OLS   Adj. R-squared:                  0.917
Method:                            Least Squares   F-statistic:                 5.991e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:42   Log-Likelihood:                 791.96
No. Observations:                           5450   AIC:                            -1580.
Df Residuals:                               5448   BIC:                            -1567.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0019      0.003     -0.561      0.575      -0.008       0.005
QQQ_Rolling_Future_Return_1y     3.0633      0.013    244.773      0.000       3.039       3.088
==============================================================================
Omnibus:                     1400.204   Durbin-Watson:                   0.045
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             6196.957
Skew:                           1.185   Prob(JB):                         0.00
Kurtosis:                       7.655   Cond. No.                         4.51
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.876
Model:                                       OLS   Adj. R-squared:                  0.876
Method:                            Least Squares   F-statistic:                 3.833e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:47   Log-Likelihood:                -3105.3
No. Observations:                           5446   AIC:                             6215.
Df Residuals:                               5444   BIC:                             6228.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1258      0.008    -15.615      0.000      -0.142      -0.110
QQQ_Rolling_Future_Return_2y     3.5394      0.018    195.792      0.000       3.504       3.575
==============================================================================
Omnibus:                     1479.731   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5094.412
Skew:                           1.346   Prob(JB):                         0.00
Kurtosis:                       6.899   Cond. No.                         3.45
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.852
Model:                                       OLS   Adj. R-squared:                  0.852
Method:                            Least Squares   F-statistic:                 3.073e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:50   Log-Likelihood:                -5069.4
No. Observations:                           5355   AIC:                         1.014e+04
Df Residuals:                               5353   BIC:                         1.016e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3905      0.014    -28.213      0.000      -0.418      -0.363
QQQ_Rolling_Future_Return_3y     3.9604      0.023    175.287      0.000       3.916       4.005
==============================================================================
Omnibus:                      617.659   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1248.527
Skew:                           0.730   Prob(JB):                    7.69e-272
Kurtosis:                       4.861   Cond. No.                         3.35
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.817
Model:                                       OLS   Adj. R-squared:                  0.817
Method:                            Least Squares   F-statistic:                 2.274e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:54   Log-Likelihood:                -7208.2
No. Observations:                           5103   AIC:                         1.442e+04
Df Residuals:                               5101   BIC:                         1.443e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.7382      0.025    -30.125      0.000      -0.786      -0.690
QQQ_Rolling_Future_Return_4y     4.5547      0.030    150.794      0.000       4.496       4.614
==============================================================================
Omnibus:                       11.531   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.003   Jarque-Bera (JB):               11.440
Skew:                          -0.104   Prob(JB):                      0.00328
Kurtosis:                       2.895   Cond. No.                         3.30
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.766
Model:                                       OLS   Adj. R-squared:                  0.766
Method:                            Least Squares   F-statistic:                 1.658e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:37:58   Log-Likelihood:                -10802.
No. Observations:                           5069   AIC:                         2.161e+04
Df Residuals:                               5067   BIC:                         2.162e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.7617      0.052    -34.121      0.000      -1.863      -1.661
QQQ_Rolling_Future_Return_5y     5.9670      0.046    128.779      0.000       5.876       6.058
==============================================================================
Omnibus:                      145.326   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              276.824
Skew:                           0.211   Prob(JB):                     7.74e-61
Kurtosis:                       4.065   Cond. No.                         3.33
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1d   R-squared:                       0.999
Model:                                       OLS   Adj. R-squared:                  0.999
Method:                            Least Squares   F-statistic:                 3.965e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:01   Log-Likelihood:                 24386.
No. Observations:                           4909   AIC:                        -4.877e+04
Df Residuals:                               4907   BIC:                        -4.875e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         7.111e-06   2.41e-05      0.296      0.768      -4e-05    5.43e-05
QQQ_Rolling_Future_Return_1d     2.9509      0.001   1991.326      0.000       2.948       2.954
==============================================================================
Omnibus:                     7547.441   Durbin-Watson:                   2.588
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         28792520.130
Skew:                          -8.741   Prob(JB):                         0.00
Kurtosis:                     377.781   Cond. No.                         61.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 8.425e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:05   Log-Likelihood:                 17077.
No. Observations:                           4909   AIC:                        -3.415e+04
Df Residuals:                               4907   BIC:                        -3.414e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0005      0.000     -4.399      0.000      -0.001      -0.000
QQQ_Rolling_Future_Return_1w     2.9538      0.003    917.893      0.000       2.947       2.960
==============================================================================
Omnibus:                     3458.321   Durbin-Watson:                   0.891
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           430383.612
Skew:                          -2.502   Prob(JB):                         0.00
Kurtosis:                      48.597   Cond. No.                         30.2
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1m   R-squared:                       0.982
Model:                                       OLS   Adj. R-squared:                  0.982
Method:                            Least Squares   F-statistic:                 2.620e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:08   Log-Likelihood:                 11004.
No. Observations:                           4909   AIC:                        -2.200e+04
Df Residuals:                               4907   BIC:                        -2.199e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0014      0.000     -3.885      0.000      -0.002      -0.001
QQQ_Rolling_Future_Return_1m     2.8967      0.006    511.819      0.000       2.886       2.908
==============================================================================
Omnibus:                     1234.803   Durbin-Watson:                   0.289
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            81528.836
Skew:                           0.182   Prob(JB):                         0.00
Kurtosis:                      22.961   Cond. No.                         15.4
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3m   R-squared:                       0.964
Model:                                       OLS   Adj. R-squared:                  0.964
Method:                            Least Squares   F-statistic:                 1.297e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:10   Log-Likelihood:                 6687.8
No. Observations:                           4909   AIC:                        -1.337e+04
Df Residuals:                               4907   BIC:                        -1.336e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0002      0.001      0.214      0.830      -0.002       0.002
QQQ_Rolling_Future_Return_3m     2.8960      0.008    360.174      0.000       2.880       2.912
==============================================================================
Omnibus:                     1438.562   Durbin-Watson:                   0.111
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            18131.784
Skew:                           1.037   Prob(JB):                         0.00
Kurtosis:                      12.184   Cond. No.                         9.10
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_6m   R-squared:                       0.940
Model:                                       OLS   Adj. R-squared:                  0.940
Method:                            Least Squares   F-statistic:                 7.744e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:12   Log-Likelihood:                 3782.9
No. Observations:                           4909   AIC:                            -7562.
Df Residuals:                               4907   BIC:                            -7549.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.002     -0.379      0.705      -0.004       0.003
QQQ_Rolling_Future_Return_6m     2.9398      0.011    278.284      0.000       2.919       2.961
==============================================================================
Omnibus:                     1092.199   Durbin-Watson:                   0.063
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5026.992
Skew:                           1.005   Prob(JB):                         0.00
Kurtosis:                       7.531   Cond. No.                         6.63
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.921
Model:                                       OLS   Adj. R-squared:                  0.921
Method:                            Least Squares   F-statistic:                 5.687e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:15   Log-Likelihood:                 835.10
No. Observations:                           4909   AIC:                            -1666.
Df Residuals:                               4907   BIC:                            -1653.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0041      0.003     -1.223      0.221      -0.011       0.002
QQQ_Rolling_Future_Return_1y     3.1266      0.013    238.483      0.000       3.101       3.152
==============================================================================
Omnibus:                     1289.790   Durbin-Watson:                   0.048
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5902.128
Skew:                           1.201   Prob(JB):                         0.00
Kurtosis:                       7.804   Cond. No.                         4.58
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.890
Model:                                       OLS   Adj. R-squared:                  0.890
Method:                            Least Squares   F-statistic:                 3.983e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:17   Log-Likelihood:                -2555.2
No. Observations:                           4909   AIC:                             5114.
Df Residuals:                               4907   BIC:                             5127.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1417      0.008    -17.613      0.000      -0.157      -0.126
QQQ_Rolling_Future_Return_2y     3.6958      0.019    199.585      0.000       3.660       3.732
==============================================================================
Omnibus:                     1260.235   Durbin-Watson:                   0.024
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4505.177
Skew:                           1.255   Prob(JB):                         0.00
Kurtosis:                       6.966   Cond. No.                         3.50
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.867
Model:                                       OLS   Adj. R-squared:                  0.867
Method:                            Least Squares   F-statistic:                 3.200e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:19   Log-Likelihood:                -4387.0
No. Observations:                           4909   AIC:                             8778.
Df Residuals:                               4907   BIC:                             8791.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4053      0.014    -29.518      0.000      -0.432      -0.378
QQQ_Rolling_Future_Return_3y     4.1032      0.023    178.878      0.000       4.058       4.148
==============================================================================
Omnibus:                      498.163   Durbin-Watson:                   0.020
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1119.255
Skew:                           0.622   Prob(JB):                    9.05e-244
Kurtosis:                       4.981   Cond. No.                         3.40
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.841
Model:                                       OLS   Adj. R-squared:                  0.841
Method:                            Least Squares   F-statistic:                 2.561e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:22   Log-Likelihood:                -6578.4
No. Observations:                           4854   AIC:                         1.316e+04
Df Residuals:                               4852   BIC:                         1.317e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.7780      0.024    -32.745      0.000      -0.825      -0.731
QQQ_Rolling_Future_Return_4y     4.7051      0.029    160.037      0.000       4.647       4.763
==============================================================================
Omnibus:                       37.794   Durbin-Watson:                   0.012
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               38.920
Skew:                          -0.203   Prob(JB):                     3.54e-09
Kurtosis:                       3.167   Cond. No.                         3.31
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.790
Model:                                       OLS   Adj. R-squared:                  0.790
Method:                            Least Squares   F-statistic:                 1.825e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:38:24   Log-Likelihood:                -10152.
No. Observations:                           4854   AIC:                         2.031e+04
Df Residuals:                               4852   BIC:                         2.032e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.8225      0.051    -36.000      0.000      -1.922      -1.723
QQQ_Rolling_Future_Return_5y     6.1390      0.045    135.095      0.000       6.050       6.228
==============================================================================
Omnibus:                      154.348   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              381.530
Skew:                           0.121   Prob(JB):                     1.42e-83
Kurtosis:                       4.352   Cond. No.                         3.32
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Rolling Returns Following Drawdowns Deviation (QQQ & TQQQ) #

rolling_returns_positive_future_returns = pd.DataFrame(index=rolling_windows.keys(), data=rolling_windows.values())
rolling_returns_positive_future_returns.reset_index(inplace=True)
rolling_returns_positive_future_returns.rename(columns={"index":"Period", 0:"Days"}, inplace=True)

for drawdown in drawdown_levels:
    temp = rolling_returns_drawdown_stats.loc[rolling_returns_drawdown_stats["Drawdown"] == drawdown]
    temp = temp[["Period", "Positive_Future_Percentage"]]
    temp.rename(columns={"Positive_Future_Percentage" : f"Positive_Future_Percentage_Post_{drawdown}_Drawdown"}, inplace=True)
    rolling_returns_positive_future_returns = pd.merge(rolling_returns_positive_future_returns, temp, left_on="Period", right_on="Period", how="outer")
    rolling_returns_positive_future_returns.sort_values(by="Days", ascending=True, inplace=True)

rolling_returns_positive_future_returns.drop(columns={"Days"}, inplace=True)
rolling_returns_positive_future_returns.reset_index(drop=True, inplace=True)
pandas_set_decimal_places(2)
display(rolling_returns_positive_future_returns.set_index("Period"))

	Positive_Future_Percentage_Post_-0.1_Drawdown	Positive_Future_Percentage_Post_-0.2_Drawdown	Positive_Future_Percentage_Post_-0.3_Drawdown	Positive_Future_Percentage_Post_-0.4_Drawdown	Positive_Future_Percentage_Post_-0.5_Drawdown	Positive_Future_Percentage_Post_-0.6_Drawdown	Positive_Future_Percentage_Post_-0.7_Drawdown	Positive_Future_Percentage_Post_-0.8_Drawdown	Positive_Future_Percentage_Post_-0.9_Drawdown
Period
1d	0.55	0.54	0.54	0.54	0.54	0.55	0.54	0.54	0.54
1w	0.56	0.56	0.56	0.56	0.56	0.57	0.57	0.56	0.56
1m	0.60	0.60	0.59	0.59	0.60	0.60	0.60	0.60	0.60
3m	0.64	0.64	0.64	0.64	0.64	0.64	0.65	0.65	0.65
6m	0.67	0.66	0.66	0.66	0.66	0.67	0.69	0.68	0.68
1y	0.71	0.71	0.71	0.71	0.71	0.71	0.73	0.74	0.73
2y	0.74	0.75	0.75	0.75	0.75	0.76	0.78	0.80	0.81
3y	0.75	0.76	0.76	0.76	0.76	0.77	0.78	0.78	0.78
4y	0.73	0.74	0.75	0.75	0.75	0.75	0.76	0.76	0.75
5y	0.73	0.74	0.75	0.75	0.75	0.75	0.76	0.76	0.76

plot_scatter(
    df=rolling_returns_positive_future_returns,
    x_plot_column="Period",
    y_plot_columns=[col for col in rolling_returns_positive_future_returns.columns if col != "Period"],
    title="TQQQ Future Return by Time Period Post Drawdown",
    x_label="Rolling Return Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Positive Future Return Percentage",
    y_format="Decimal",
    y_format_decimal_places=2,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

This plot summarizes the future rolling returns well. For rolling returns up to ~3 months following all drawdown levels, we see the rolling returns of TQQQ are positive ~65% of the time.

As we extend the time horizon, the percentage of positive rolling returns increases, which is consistent with the idea that the longer you hold through and post drawdown, the more likely you are to recover and achieve positive returns.

From a timing standpoint, this analysis suggests that the optimal time to buy TQQQ would be following a drawdown of 70% or more, and holding for at least 3 years. The data tells us that having a positive rolling return over time is ~75%.

One might consider the idea of allocating to TQQQ via a ladder, starting at a drawdown of 50%, and continuing to add to the position as the drawdown deepens, with the idea that the more severe the drawdown, the higher the expected future returns. However, this strategy could require enduring significant volatility, as one would be adding to the position during periods of paper losses.

SPY & UPRO #

Next, we will repeat the same analysis for SPY and UPRO, and see how the results compare to those of QQQ and TQQQ.

Acquire & Plot Data (SPY) #

First, let’s get the data for SPY. If we already have the desired data, we can load it from a local pickle file. Otherwise, we can download it from Yahoo Finance using the yf_pull_data function.

pandas_set_decimal_places(2)

yf_pull_data(
    base_directory=DATA_DIR,
    ticker="SPY",
    adjusted=False,
    source="Yahoo_Finance",
    asset_class="Exchange_Traded_Funds",
    excel_export=True,
    pickle_export=True,
    output_confirmation=False,
)

spy = load_data(
    base_directory=DATA_DIR,
    ticker="SPY",
    source="Yahoo_Finance",
    asset_class="Exchange_Traded_Funds",
    timeframe="Daily",
    file_format="pickle",
)

# Rename columns to "SPY_Close", etc.
spy = spy.rename(columns={
    "Adj Close": "SPY_Adj_Close",
    "Close": "SPY_Close",
    "High": "SPY_High",
    "Low": "SPY_Low",
    "Open": "SPY_Open",
    "Volume": "SPY_Volume"
})

display(spy)

	SPY_Adj_Close	SPY_Close	SPY_High	SPY_Low	SPY_Open	SPY_Volume
Date
1993-01-29	24.18	43.94	43.97	43.75	43.97	1003200
1993-02-01	24.35	44.25	44.25	43.97	43.97	480500
1993-02-02	24.40	44.34	44.38	44.12	44.22	201300
1993-02-03	24.66	44.81	44.84	44.38	44.41	529400
1993-02-04	24.76	45.00	45.09	44.47	44.97	531500
...	...	...	...	...	...	...
2026-06-10	725.43	725.43	738.38	725.33	733.39	60341300
2026-06-11	737.76	737.76	740.00	724.41	728.76	86330500
2026-06-12	741.75	741.75	744.44	735.03	740.71	57079500
2026-06-15	754.83	754.83	756.68	751.76	751.85	60176400
2026-06-16	750.33	750.33	755.44	749.88	754.55	67093100

8402 rows × 6 columns

And the plot of the time series of partially adjusted close prices:

plot_time_series(
    df=spy,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Adj_Close"],
    title="SPY Adjusted Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

Acquire & Plot Data (UPRO) #

Next, UPRO:

yf_pull_data(
    base_directory=DATA_DIR,
    ticker="UPRO",
    adjusted=False,
    source="Yahoo_Finance", 
    asset_class="Exchange_Traded_Funds", 
    excel_export=True,
    pickle_export=True,
    output_confirmation=False,
)
    
upro = load_data(
    base_directory=DATA_DIR,
    ticker="UPRO",
    source="Yahoo_Finance", 
    asset_class="Exchange_Traded_Funds",
    timeframe="Daily",
    file_format="pickle",
)

# Rename columns to "UPRO_Close", etc.
upro = upro.rename(columns={
    "Adj Close": "UPRO_Adj_Close",
    "Close": "UPRO_Close", 
    "High": "UPRO_High", 
    "Low": "UPRO_Low", 
    "Open": "UPRO_Open", 
    "Volume": "UPRO_Volume"
})

display(upro)

	UPRO_Adj_Close	UPRO_Close	UPRO_High	UPRO_Low	UPRO_Open	UPRO_Volume
Date
2009-06-25	1.13	1.21	1.21	1.13	1.13	2577600
2009-06-26	1.13	1.20	1.21	1.18	1.20	13104000
2009-06-29	1.16	1.23	1.24	1.19	1.21	8690400
2009-06-30	1.13	1.20	1.24	1.18	1.23	17128800
2009-07-01	1.14	1.22	1.25	1.21	1.22	12038400
...	...	...	...	...	...	...
2026-06-10	130.69	130.69	137.97	130.68	135.18	4295800
2026-06-11	137.29	137.29	138.58	130.15	132.43	3955000
2026-06-12	139.41	139.41	140.95	135.75	138.87	2894200
2026-06-15	146.74	146.74	147.87	145.10	145.14	2673100
2026-06-16	144.14	144.14	147.11	143.88	146.64	1787300

4270 rows × 6 columns

And the plot of the time series of partially adjusted close prices:

plot_time_series(
    df=upro,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["UPRO_Adj_Close"],
    title="UPRO Adjusted Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

Calculate & Plot Cumulative Returns, Rolling Returns, and Drawdowns (SPY & UPRO) #

Next, we will calculate the cumulative returns, rolling returns, and drawdowns. This involves aligning the data to start with the inception of UPRO. For this excercise, we will not extrapolate the data for SPY back to 1993, but rather just align the data from the inception of UPRO in 2009.

etfs = ["SPY", "UPRO"]

# Merge dataframes and drop rows with missing values
spy_upro_aligned = upro.merge(spy, left_index=True, right_index=True, how='left')
spy_upro_aligned = spy_upro_aligned.dropna()

# Calculate cumulative returns
for etf in etfs:
    spy_upro_aligned[f"{etf}_Return"] = spy_upro_aligned[f"{etf}_Close"].pct_change()
    spy_upro_aligned[f"{etf}_Cumulative_Return"] = (1 + spy_upro_aligned[f"{etf}_Return"]).cumprod() - 1
    spy_upro_aligned[f"{etf}_Cumulative_Return_Plus_One"] = 1 + spy_upro_aligned[f"{etf}_Cumulative_Return"]
    spy_upro_aligned[f"{etf}_Rolling_Max"] = spy_upro_aligned[f"{etf}_Cumulative_Return_Plus_One"].cummax()
    spy_upro_aligned[f"{etf}_Drawdown"] = spy_upro_aligned[f"{etf}_Cumulative_Return_Plus_One"] / spy_upro_aligned[f"{etf}_Rolling_Max"] - 1
    spy_upro_aligned.drop(columns=[f"{etf}_Cumulative_Return_Plus_One", f"{etf}_Rolling_Max"], inplace=True)

# Define rolling windows in trading days
rolling_windows = {
    '1d': 1,      # 1 day
    '1w': 5,      # 1 week (5 trading days)
    '1m': 21,     # 1 month (~21 trading days)
    '3m': 63,     # 3 months (~63 trading days)
    '6m': 126,    # 6 months (~126 trading days)
    '1y': 252,    # 1 year (~252 trading days)
    '2y': 504,    # 2 years (~504 trading days)
    '3y': 756,    # 3 years (~756 trading days)
    '4y': 1008,   # 4 years (~1008 trading days)
    '5y': 1260,   # 5 years (~1260 trading days)
}

# Calculate rolling returns for each ETF and each window
for etf in etfs:
    for period_name, window in rolling_windows.items():
        spy_upro_aligned[f"{etf}_Rolling_Return_{period_name}"] = (
            spy_upro_aligned[f"{etf}_Close"].pct_change(periods=window)
        )
        
display(spy_upro_aligned)

	UPRO_Adj_Close	UPRO_Close	UPRO_High	UPRO_Low	UPRO_Open	UPRO_Volume	SPY_Adj_Close	SPY_Close	SPY_High	SPY_Low	...	UPRO_Rolling_Return_1d	UPRO_Rolling_Return_1w	UPRO_Rolling_Return_1m	UPRO_Rolling_Return_3m	UPRO_Rolling_Return_6m	UPRO_Rolling_Return_1y	UPRO_Rolling_Return_2y	UPRO_Rolling_Return_3y	UPRO_Rolling_Return_4y	UPRO_Rolling_Return_5y
Date
2009-06-25	1.13	1.21	1.21	1.13	1.13	2577600	68.20	92.08	92.17	89.57	...	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2009-06-26	1.13	1.20	1.21	1.18	1.20	13104000	68.03	91.84	92.24	91.27	...	-0.01	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2009-06-29	1.16	1.23	1.24	1.19	1.21	8690400	68.66	92.70	92.82	91.60	...	0.03	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2009-06-30	1.13	1.20	1.24	1.18	1.23	17128800	68.11	91.95	93.06	91.27	...	-0.02	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
2009-07-01	1.14	1.22	1.25	1.21	1.22	12038400	68.39	92.33	93.23	92.21	...	0.01	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2026-06-10	130.69	130.69	137.97	130.68	135.18	4295800	725.43	725.43	738.38	725.33	...	-0.05	-0.12	-0.07	0.19	0.12	0.56	0.79	2.08	1.73	1.48
2026-06-11	137.29	137.29	138.58	130.15	132.43	3955000	737.76	737.76	740.00	724.41	...	0.05	-0.08	-0.02	0.31	0.18	0.61	0.88	2.21	2.01	1.54
2026-06-12	139.41	139.41	140.95	135.75	138.87	2894200	741.75	741.75	744.44	735.03	...	0.02	0.01	-0.02	0.35	0.17	0.65	0.92	2.29	2.04	1.58
2026-06-15	146.74	146.74	147.87	145.10	145.14	2673100	754.83	754.83	756.68	751.76	...	0.05	0.06	0.01	0.38	0.23	0.72	1.00	2.41	2.11	1.72
2026-06-16	144.14	144.14	147.11	143.88	146.64	1787300	750.33	750.33	755.44	749.88	...	-0.02	0.05	0.03	0.35	0.25	0.75	0.95	2.33	2.15	1.68

4270 rows × 38 columns

And now the plot for the cumulative returns:

plot_time_series(
    df=spy_upro_aligned,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Cumulative_Return", "UPRO_Cumulative_Return"],
    title="Cumulative Returns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Cumulative Return",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

And the drawdown plot:

plot_time_series(
    df=spy_upro_aligned,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Drawdown", "UPRO_Drawdown"],
    title="Drawdowns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Drawdown",
    y_format="Percentage",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

Summary Statistics (SPY & UPRO) #

Looking at the summary statistics further confirms our intuitions about the volatility and drawdowns.

spy_sum_stats = summary_stats(
    fund_list=["SPY"],
    df=spy_upro_aligned[["SPY_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

upro_sum_stats = summary_stats(
    fund_list=["UPRO"],
    df=spy_upro_aligned[["UPRO_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

sum_stats = pd.concat([spy_sum_stats, upro_sum_stats])

display(sum_stats)

	Annual Mean Return (Arithmetic)	Annualized Volatility	Annualized Sharpe Ratio	CAGR (Geometric)	Daily Max Return	Daily Max Return (Date)	Daily Min Return	Daily Min Return (Date)	Max Drawdown	Peak	Trough	Recovery Date	Calendar Days to Recovery	MAR Ratio
SPY_Return	0.14	0.17	0.81	0.13	0.11	2025-04-09	-0.11	2020-03-16	-0.34	2020-02-19	2020-03-23	2020-08-18	148	0.39
UPRO_Return	0.42	0.51	0.81	0.33	0.28	2020-03-24	-0.35	2020-03-16	-0.77	2020-02-19	2020-03-23	2021-01-08	291	0.42

Plot Returns & Verify Beta (SPY & UPRO) #

Before we look at the rolling returns, let us first verify that the daily returns for UPRO are in fact ~3x those of SPY.

plot_scatter(
    df=spy_upro_aligned,
    x_plot_column="SPY_Return",
    y_plot_columns=["UPRO_Return"],
    title="SPY & UPRO Returns",
    x_label="SPY Return",
    x_format="Decimal",
    x_format_decimal_places=2,
    x_tick_spacing="Auto",
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="UPRO Return",
    y_format="Decimal",
    y_format_decimal_places=2,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=True,
    OLS_column="UPRO_Return",
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=True,
    RidgeCV_column="UPRO_Return",
    regression_constant=True,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

model = run_regression(
    df=spy_upro_aligned,
    x_plot_column="SPY_Return",
    y_plot_column="UPRO_Return",
    regression_model="OLS-statsmodels",
    regression_constant=True,
)

print(model.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:            UPRO_Return   R-squared:                       0.994
Model:                            OLS   Adj. R-squared:                  0.994
Method:                 Least Squares   F-statistic:                 6.917e+05
Date:                Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                        14:38:31   Log-Likelihood:                 19473.
No. Observations:                4269   AIC:                        -3.894e+04
Df Residuals:                    4267   BIC:                        -3.893e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const       1.289e-05   3.87e-05      0.333      0.739   -6.31e-05    8.88e-05
SPY_Return     2.9758      0.004    831.659      0.000       2.969       2.983
==============================================================================
Omnibus:                     2736.798   Durbin-Watson:                   2.589
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           536793.027
Skew:                           2.001   Prob(JB):                         0.00
Kurtosis:                      57.789   Cond. No.                         92.5
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Similar to QQQ/TQQQ, this plot makes sense and we can see that there is a strong clustering of points, but we double check with the regression, regressing the UPRO daily return (y) on the SPY daily return (X).

Extrapolate Data (SPY & UPRO) #

We will now extrapolate the returns of SPY to backfill the data from the inception of SPY in 1993 to the inception of UPRO in 2009. For this, we’ll use the coefficient of 2.98 that we found in the regression results above.

# Set leverage multiplier based on regression coefficient
LEVERAGE_MULTIPLIER = model.params.iloc[1]

# Merge dataframes and extrapolate return values for SPY back to 1993 using the leverage multiplier
spy_upro_extrap = spy[["SPY_Close"]].merge(upro[["UPRO_Close"]], left_index=True, right_index=True, how='left')

etfs = ["SPY", "UPRO"]

# Calculate cumulative returns
for etf in etfs:
    spy_upro_extrap[f"{etf}_Return"] = spy_upro_extrap[f"{etf}_Close"].pct_change()

# Extrapolate UPRO returns for missing values
spy_upro_extrap["UPRO_Return"] = spy_upro_extrap["UPRO_Return"].fillna(LEVERAGE_MULTIPLIER * spy_upro_extrap["SPY_Return"])

# Find the first valid UPRO_Close index and value
first_valid_idx = spy_upro_extrap['UPRO_Close'].first_valid_index()
print(first_valid_idx)
first_valid_price = spy_upro_extrap.loc[first_valid_idx, 'UPRO_Close']
print(first_valid_price)

2009-06-25 00:00:00
1.205556035041809

Before we extrapolate, let’s first look at the data we have for SPY and UPRO around the inception of UPRO in 2009:

# Check values around the first valid index
pandas_set_decimal_places(4)
display(spy_upro_extrap.loc["2009-06-20":"2009-06-30"])

	SPY_Close	UPRO_Close	SPY_Return	UPRO_Return
Date
2009-06-22	89.2800	NaN	-0.0300	-0.0892
2009-06-23	89.3500	NaN	0.0008	0.0023
2009-06-24	90.1200	NaN	0.0086	0.0256
2009-06-25	92.0800	1.2056	0.0217	0.0647
2009-06-26	91.8400	1.1993	-0.0026	-0.0052
2009-06-29	92.7000	1.2333	0.0094	0.0284
2009-06-30	91.9500	1.2039	-0.0081	-0.0239

Now, backfill the data for the UPRO close price:

# Iterate through the dataframe backwards
for i in range(spy_upro_extrap.index.get_loc(first_valid_idx) - 1, -1, -1):
    
    # The return that led to the price the next day
    current_return = spy_upro_extrap.iloc[i + 1]['UPRO_Return']

    # Get the next day's price
    next_price = spy_upro_extrap.iloc[i + 1]['UPRO_Close']
    
    # Price_{t} = Price_{t+1} / (1 + Return_{t})
    spy_upro_extrap.loc[spy_upro_extrap.index[i], 'UPRO_Close'] = next_price / (1 + current_return)

Finally, confirm the values are correct:

# Confirm values around the first valid index after extrapolation
display(spy_upro_extrap.loc["2009-06-20":"2009-06-30"])

	SPY_Close	UPRO_Close	SPY_Return	UPRO_Return
Date
2009-06-22	89.2800	1.1014	-0.0300	-0.0892
2009-06-23	89.3500	1.1040	0.0008	0.0023
2009-06-24	90.1200	1.1323	0.0086	0.0256
2009-06-25	92.0800	1.2056	0.0217	0.0647
2009-06-26	91.8400	1.1993	-0.0026	-0.0052
2009-06-29	92.7000	1.2333	0.0094	0.0284
2009-06-30	91.9500	1.2039	-0.0081	-0.0239

And the complete DataFrame with the extrapolated values:

pandas_set_decimal_places(2)
display(spy_upro_extrap)

	SPY_Close	UPRO_Close	SPY_Return	UPRO_Return
Date
1993-01-29	43.94	0.93	NaN	NaN
1993-02-01	44.25	0.95	0.01	0.02
1993-02-02	44.34	0.95	0.00	0.01
1993-02-03	44.81	0.98	0.01	0.03
1993-02-04	45.00	0.99	0.00	0.01
...	...	...	...	...
2026-06-10	725.43	130.69	-0.02	-0.05
2026-06-11	737.76	137.29	0.02	0.05
2026-06-12	741.75	139.41	0.01	0.02
2026-06-15	754.83	146.74	0.02	0.05
2026-06-16	750.33	144.14	-0.01	-0.02

8402 rows × 4 columns

After the extrapolation, we now have the following plots for the prices, cumulative returns, and drawdowns:

etfs = ["SPY", "UPRO"]

# Calculate cumulative returns
for etf in etfs:
    spy_upro_extrap[f"{etf}_Return"] = spy_upro_extrap[f"{etf}_Close"].pct_change()
    spy_upro_extrap[f"{etf}_Cumulative_Return"] = (1 + spy_upro_extrap[f"{etf}_Return"]).cumprod() - 1
    spy_upro_extrap[f"{etf}_Cumulative_Return_Plus_One"] = 1 + spy_upro_extrap[f"{etf}_Cumulative_Return"]
    spy_upro_extrap[f"{etf}_Rolling_Max"] = spy_upro_extrap[f"{etf}_Cumulative_Return_Plus_One"].cummax()
    spy_upro_extrap[f"{etf}_Drawdown"] = spy_upro_extrap[f"{etf}_Cumulative_Return_Plus_One"] / spy_upro_extrap[f"{etf}_Rolling_Max"] - 1
    spy_upro_extrap.drop(columns=[f"{etf}_Cumulative_Return_Plus_One", f"{etf}_Rolling_Max"], inplace=True)

plot_time_series(
    df=spy_upro_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Close"],
    title="SPY Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=spy_upro_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["UPRO_Close"],
    title="UPRO Close Price",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Price ($)",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=False,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=spy_upro_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Cumulative_Return", "UPRO_Cumulative_Return"],
    title="Cumulative Returns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=2,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Cumulative Return",
    y_format="Decimal",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_time_series(
    df=spy_upro_extrap,
    plot_start_date=None,
    plot_end_date=None,
    plot_columns=["SPY_Drawdown", "UPRO_Drawdown"],
    title="Drawdowns",
    x_label="Date",
    x_format="Year",
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=30,
    y_label="Drawdown",
    y_format="Percentage",
    y_format_decimal_places=0,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

spy_extrap_sum_stats = summary_stats(
    fund_list=["SPY"],
    df=spy_upro_extrap[["SPY_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

upro_extrap_sum_stats = summary_stats(
    fund_list=["UPRO"],
    df=spy_upro_extrap[["UPRO_Return"]],
    period="Daily",
    use_calendar_days=False,
    excel_export=False,
    pickle_export=False,
    output_confirmation=False,
)

sum_stats = pd.concat([spy_sum_stats, upro_sum_stats, spy_extrap_sum_stats, upro_extrap_sum_stats])
sum_stats.index = ["SPY (2009 - Present)", "UPRO (2009 - Present)", "SPY (1993 - Present)", "UPRO Extrapolated (1993 - Present)"]

display(sum_stats)

	Annual Mean Return (Arithmetic)	Annualized Volatility	Annualized Sharpe Ratio	CAGR (Geometric)	Daily Max Return	Daily Max Return (Date)	Daily Min Return	Daily Min Return (Date)	Max Drawdown	Peak	Trough	Recovery Date	Calendar Days to Recovery	MAR Ratio
SPY (2009 - Present)	0.14	0.17	0.81	0.13	0.11	2025-04-09	-0.11	2020-03-16	-0.34	2020-02-19	2020-03-23	2020-08-18	148	0.39
UPRO (2009 - Present)	0.42	0.51	0.81	0.33	0.28	2020-03-24	-0.35	2020-03-16	-0.77	2020-02-19	2020-03-23	2021-01-08	291	0.42
SPY (1993 - Present)	0.10	0.19	0.55	0.09	0.15	2008-10-13	-0.11	2020-03-16	-0.56	2007-10-09	2009-03-09	2013-03-14	1466	0.16
UPRO Extrapolated (1993 - Present)	0.31	0.55	0.55	0.16	0.43	2008-10-13	-0.35	2020-03-16	-0.98	2000-03-24	2009-03-09	2017-11-30	3188	0.17

Interestingly, the maximum drawdown for UPRO is not as severe as that of TQQQ, which may be due to that SPY has not had the same extreme return profile as QQQ. This highlights the importance of the underlying asset’s return profile on the performance of leveraged ETFs.

Plot Rolling Returns (SPY & UPRO) #

Next, we will consider the following:

Histogram and scatter plots of the rolling returns of SPY and UPRO
Regressions to establish a “leverage factor” for the rolling returns
The deviation from a 3x return for each time period

For this set of regressions, we will also allow the constant. First, we need the rolling returns for various time periods:

# Define rolling windows in trading days
rolling_windows = {
    '1d': 1,      # 1 day
    '1w': 5,      # 1 week (5 trading days)
    '1m': 21,     # 1 month (~21 trading days)
    '3m': 63,     # 3 months (~63 trading days)
    '6m': 126,    # 6 months (~126 trading days)
    '1y': 252,    # 1 year (~252 trading days)
    '2y': 504,    # 2 years (~504 trading days)
    '3y': 756,    # 3 years (~756 trading days)
    '4y': 1008,   # 4 years (~1008 trading days)
    '5y': 1260,   # 5 years (~1260 trading days)
}

# Calculate rolling returns for each ETF and each window
for etf in etfs:
    for period_name, window in rolling_windows.items():
        spy_upro_extrap[f"{etf}_Rolling_Return_{period_name}"] = (
            spy_upro_extrap[f"{etf}_Close"].pct_change(periods=window)
        )

This gives us the following series of histograms, scatter plots, and regression model results:

# Create a dataframe to hold rolling returns stats
rolling_returns_stats = pd.DataFrame()

for period_name, window in rolling_windows.items():
    plot_histogram(
        df=spy_upro_extrap,
        plot_columns=[f"SPY_Rolling_Return_{period_name}", f"UPRO_Rolling_Return_{period_name}"],
        title=f"SPY & UPRO {period_name} Rolling Returns",
        x_label="Rolling Return",
        x_tick_spacing="Auto",
        x_tick_rotation=30,
        y_label="# Of Datapoints",
        y_tick_spacing="Auto",
        y_tick_rotation=0,
        grid=True,
        legend=True,
        export_plot=False,
        plot_file_name=None,
    )

    plot_scatter(
        df=spy_upro_extrap,
        x_plot_column=f"SPY_Rolling_Return_{period_name}",
        y_plot_columns=[f"UPRO_Rolling_Return_{period_name}"],
        title=f"SPY & UPRO {period_name} Rolling Returns",
        x_label="SPY Rolling Return",
        x_format="Decimal",
        x_format_decimal_places=2,
        x_tick_spacing="Auto",
        x_tick_start=None,
        x_tick_rotation=30,
        y_label="UPRO Rolling Return",
        y_format="Decimal",
        y_format_decimal_places=2,
        y_tick_spacing="Auto",
        y_tick_rotation=0,
        plot_OLS_regression_line=True,
        OLS_column=f"UPRO_Rolling_Return_{period_name}",
        plot_Ridge_regression_line=False,
        Ridge_column=None,
        plot_RidgeCV_regression_line=True,
        RidgeCV_column=f"UPRO_Rolling_Return_{period_name}",
        regression_constant=True,
        grid=True,
        legend=True,
        export_plot=False,
        plot_file_name=None,
    )

    # Run OLS regression with statsmodels
    model = run_regression(
        df=spy_upro_extrap,
        x_plot_column=f"SPY_Rolling_Return_{period_name}",
        y_plot_column=f"UPRO_Rolling_Return_{period_name}",
        regression_model="OLS-statsmodels",
        regression_constant=True,
    )
    print(model.summary())

    # Add the regression results to the rolling returns stats dataframe
    intercept = model.params.iloc[0]
    intercept_pvalue = model.pvalues.iloc[0]   # p-value for Intercept
    slope = model.params.iloc[1]
    slope_pvalue = model.pvalues.iloc[1]       # p-value for SPY_Return
    r_squared = model.rsquared

    # Calc skew
    return_ratio = spy_upro_extrap[f'UPRO_Rolling_Return_{period_name}'] / spy_upro_extrap[f'SPY_Rolling_Return_{period_name}']
    skew = return_ratio.skew()

    # Calc conditional symmetry
    up_markets = spy_upro_extrap[spy_upro_extrap[f'SPY_Rolling_Return_{period_name}'] > 0]
    down_markets = spy_upro_extrap[spy_upro_extrap[f'SPY_Rolling_Return_{period_name}'] <= 0]

    avg_beta_up = (up_markets[f'UPRO_Rolling_Return_{period_name}'] / up_markets[f'SPY_Rolling_Return_{period_name}']).mean()
    avg_beta_down = (down_markets[f'UPRO_Rolling_Return_{period_name}'] / down_markets[f'SPY_Rolling_Return_{period_name}']).mean()

    asymmetry = avg_beta_up - avg_beta_down

    rolling_returns_slope_int = pd.DataFrame({
        "Period": period_name,
        "Intercept": [intercept],
        # "Intercept_PValue": [intercept_pvalue],
        "Slope": [slope],
        # "Slope_PValue": [slope_pvalue],
        "R_Squared": [r_squared],
        "Skew": [skew],
        "Average Upside Beta": [avg_beta_up],
        "Average Downside Beta": [avg_beta_down],
        "Asymmetry": [asymmetry]
    })

    rolling_returns_stats = pd.concat([rolling_returns_stats, rolling_returns_slope_int])

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_1d   R-squared:                       0.997
Model:                                OLS   Adj. R-squared:                  0.997
Method:                     Least Squares   F-statistic:                 3.150e+06
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:39   Log-Likelihood:                 41165.
No. Observations:                    8401   AIC:                        -8.233e+04
Df Residuals:                        8399   BIC:                        -8.231e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                  6.545e-06   1.97e-05      0.333      0.739    -3.2e-05    4.51e-05
SPY_Rolling_Return_1d     2.9759      0.002   1774.746      0.000       2.973       2.979
==============================================================================
Omnibus:                     6935.216   Durbin-Watson:                   2.589
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          4305371.385
Skew:                           2.818   Prob(JB):                         0.00
Kurtosis:                     113.760   Cond. No.                         85.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_1w   R-squared:                       0.994
Model:                                OLS   Adj. R-squared:                  0.994
Method:                     Least Squares   F-statistic:                 1.422e+06
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:41   Log-Likelihood:                 31865.
No. Observations:                    8397   AIC:                        -6.373e+04
Df Residuals:                        8395   BIC:                        -6.371e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0003   5.96e-05     -4.437      0.000      -0.000      -0.000
SPY_Rolling_Return_1w     2.9726      0.002   1192.595      0.000       2.968       2.977
==============================================================================
Omnibus:                     3767.042   Durbin-Watson:                   0.954
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1513474.088
Skew:                          -0.842   Prob(JB):                         0.00
Kurtosis:                      68.749   Cond. No.                         42.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_1m   R-squared:                       0.988
Model:                                OLS   Adj. R-squared:                  0.988
Method:                     Least Squares   F-statistic:                 6.771e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:44   Log-Likelihood:                 23325.
No. Observations:                    8381   AIC:                        -4.665e+04
Df Residuals:                        8379   BIC:                        -4.663e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0015      0.000     -9.082      0.000      -0.002      -0.001
SPY_Rolling_Return_1m     2.9618      0.004    822.882      0.000       2.955       2.969
==============================================================================
Omnibus:                     2889.453   Durbin-Watson:                   0.313
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           869076.673
Skew:                          -0.269   Prob(JB):                         0.00
Kurtosis:                      52.884   Cond. No.                         22.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_3m   R-squared:                       0.979
Model:                                OLS   Adj. R-squared:                  0.979
Method:                     Least Squares   F-statistic:                 3.867e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:46   Log-Likelihood:                 16567.
No. Observations:                    8339   AIC:                        -3.313e+04
Df Residuals:                        8337   BIC:                        -3.312e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0070      0.000    -18.201      0.000      -0.008      -0.006
SPY_Rolling_Return_3m     3.0477      0.005    621.860      0.000       3.038       3.057
==============================================================================
Omnibus:                     2443.395   Durbin-Watson:                   0.136
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           134196.605
Skew:                           0.591   Prob(JB):                         0.00
Kurtosis:                      22.617   Cond. No.                         13.5
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_6m   R-squared:                       0.957
Model:                                OLS   Adj. R-squared:                  0.957
Method:                     Least Squares   F-statistic:                 1.839e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:49   Log-Likelihood:                 10257.
No. Observations:                    8276   AIC:                        -2.051e+04
Df Residuals:                        8274   BIC:                        -2.050e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0113      0.001    -13.412      0.000      -0.013      -0.010
SPY_Rolling_Return_6m     3.0704      0.007    428.867      0.000       3.056       3.084
==============================================================================
Omnibus:                     2098.788   Durbin-Watson:                   0.055
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            26735.476
Skew:                           0.854   Prob(JB):                         0.00
Kurtosis:                      11.638   Cond. No.                         9.32
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_1y   R-squared:                       0.927
Model:                                OLS   Adj. R-squared:                  0.927
Method:                     Least Squares   F-statistic:                 1.036e+05
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:52   Log-Likelihood:                 3981.9
No. Observations:                    8150   AIC:                            -7960.
Df Residuals:                        8148   BIC:                            -7946.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0199      0.002    -10.334      0.000      -0.024      -0.016
SPY_Rolling_Return_1y     3.2103      0.010    321.847      0.000       3.191       3.230
==============================================================================
Omnibus:                     1417.642   Durbin-Watson:                   0.031
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             6684.393
Skew:                           0.770   Prob(JB):                         0.00
Kurtosis:                       7.161   Cond. No.                         6.13
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_2y   R-squared:                       0.895
Model:                                OLS   Adj. R-squared:                  0.895
Method:                     Least Squares   F-statistic:                 6.755e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:55   Log-Likelihood:                -2159.8
No. Observations:                    7898   AIC:                             4324.
Df Residuals:                        7896   BIC:                             4338.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0530      0.005    -11.481      0.000      -0.062      -0.044
SPY_Rolling_Return_2y     3.5538      0.014    259.906      0.000       3.527       3.581
==============================================================================
Omnibus:                      968.075   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1515.839
Skew:                           0.873   Prob(JB):                         0.00
Kurtosis:                       4.248   Cond. No.                         4.01
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_3y   R-squared:                       0.866
Model:                                OLS   Adj. R-squared:                  0.866
Method:                     Least Squares   F-statistic:                 4.922e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:38:58   Log-Likelihood:                -7086.8
No. Observations:                    7646   AIC:                         1.418e+04
Df Residuals:                        7644   BIC:                         1.419e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.2353      0.009    -24.772      0.000      -0.254      -0.217
SPY_Rolling_Return_3y     4.3732      0.020    221.846      0.000       4.335       4.412
==============================================================================
Omnibus:                     1346.677   Durbin-Watson:                   0.008
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2615.290
Skew:                           1.077   Prob(JB):                         0.00
Kurtosis:                       4.889   Cond. No.                         3.16
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_4y   R-squared:                       0.860
Model:                                OLS   Adj. R-squared:                  0.860
Method:                     Least Squares   F-statistic:                 4.527e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:39:01   Log-Likelihood:                -10394.
No. Observations:                    7394   AIC:                         2.079e+04
Df Residuals:                        7392   BIC:                         2.081e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.5564      0.016    -34.922      0.000      -0.588      -0.525
SPY_Rolling_Return_4y     5.3537      0.025    212.761      0.000       5.304       5.403
==============================================================================
Omnibus:                     1112.745   Durbin-Watson:                   0.008
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2314.886
Skew:                           0.909   Prob(JB):                         0.00
Kurtosis:                       5.052   Cond. No.                         2.70
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     UPRO_Rolling_Return_5y   R-squared:                       0.848
Model:                                OLS   Adj. R-squared:                  0.848
Method:                     Least Squares   F-statistic:                 3.973e+04
Date:                    Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                            14:39:04   Log-Likelihood:                -12924.
No. Observations:                    7142   AIC:                         2.585e+04
Df Residuals:                        7140   BIC:                         2.587e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -1.0162      0.025    -41.008      0.000      -1.065      -0.968
SPY_Rolling_Return_5y     6.2683      0.031    199.327      0.000       6.207       6.330
==============================================================================
Omnibus:                      700.093   Durbin-Watson:                   0.007
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1322.133
Skew:                           0.660   Prob(JB):                    7.99e-288
Kurtosis:                       4.644   Cond. No.                         2.52
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Rolling Returns Deviation (SPY & UPRO) #

rolling_returns_stats["Return_Deviation_From_3x"] = rolling_returns_stats["Slope"] - 3.0
pandas_set_decimal_places(3)
display(rolling_returns_stats.set_index("Period"))

	Intercept	Slope	R_Squared	Skew	Average Upside Beta	Average Downside Beta	Asymmetry	Return_Deviation_From_3x
Period
1d	0.000	2.976	0.997	NaN	2.938	NaN	NaN	-0.024
1w	-0.000	2.973	0.994	NaN	2.756	NaN	NaN	-0.027
1m	-0.002	2.962	0.988	NaN	2.496	-inf	inf	-0.038
3m	-0.007	3.048	0.979	NaN	2.009	-inf	inf	0.048
6m	-0.011	3.070	0.957	NaN	1.029	-inf	inf	0.070
1y	-0.020	3.210	0.927	NaN	1.651	-inf	inf	0.210
2y	-0.053	3.554	0.895	0.351	1.866	9.298	-7.432	0.554
3y	-0.235	4.373	0.866	-6.094	1.594	8.620	-7.026	1.373
4y	-0.556	5.354	0.860	-66.216	0.045	7.234	-7.190	2.354
5y	-1.016	6.268	0.848	-35.378	-2.209	21.002	-23.210	3.268

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Return_Deviation_From_3x"],
    title="UPRO Deviation from Perfect 3x Leverage by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Deviation from 3x Leverage",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Slope"],
    title="UPRO Slope by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Slope",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

plot_scatter(
    df=rolling_returns_stats,
    x_plot_column="Period",
    y_plot_columns=["Intercept"],
    title="Intercept by Time Period",
    x_label="Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Intercept",
    y_format="Decimal",
    y_format_decimal_places=1,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

display(rolling_returns_stats.set_index("Period"))

	Intercept	Slope	R_Squared	Skew	Average Upside Beta	Average Downside Beta	Asymmetry	Return_Deviation_From_3x
Period
1d	0.000	2.976	0.997	NaN	2.938	NaN	NaN	-0.024
1w	-0.000	2.973	0.994	NaN	2.756	NaN	NaN	-0.027
1m	-0.002	2.962	0.988	NaN	2.496	-inf	inf	-0.038
3m	-0.007	3.048	0.979	NaN	2.009	-inf	inf	0.048
6m	-0.011	3.070	0.957	NaN	1.029	-inf	inf	0.070
1y	-0.020	3.210	0.927	NaN	1.651	-inf	inf	0.210
2y	-0.053	3.554	0.895	0.351	1.866	9.298	-7.432	0.554
3y	-0.235	4.373	0.866	-6.094	1.594	8.620	-7.026	1.373
4y	-0.556	5.354	0.860	-66.216	0.045	7.234	-7.190	2.354
5y	-1.016	6.268	0.848	-35.378	-2.209	21.002	-23.210	3.268

Similar as to QQQ/TQQQ, up to 1 year, there is minimal difference between the mean UPRO 1 year rolling return and the hypothetical 3x leverage, with an R^2 of greater than 0.9.

However, as we extend the time period, we see that

The “leverage factor” increases significantly, resulting in a deviation from the perfect 3x leverage.
The intercept also begins to deviate significantly from 0.

Rolling Returns Following Drawdowns (SPY & UPRO) #

We will identify the drawdown levels of UPRO and then look at the subsequent rolling returns over various time horizons.

# Copy DataFrame
spy_upro_extrap_future = spy_upro_extrap.copy()

# Create a list of drawdown levels to analyze
drawdown_levels = [-0.10, -0.20, -0.30, -0.40, -0.50, -0.60, -0.70, -0.80, -0.90]

# Shift the rolling return columns by the number of days in the rolling window to get the returns following the drawdown
for etf in etfs:
    for period_name, window in rolling_windows.items():
        spy_upro_extrap_future[f"{etf}_Rolling_Future_Return_{period_name}"] = spy_upro_extrap_future[f"{etf}_Rolling_Return_{period_name}"].shift(-window)

Now, we can analyze the future rolling returns following specific drawdown levels:

# Create a dataframe to hold rolling returns stats
rolling_returns_drawdown_stats = pd.DataFrame()

for drawdown in drawdown_levels:

    for period_name, window in rolling_windows.items():

        try:
            plot_histogram(
                df=spy_upro_extrap_future[spy_upro_extrap_future["UPRO_Drawdown"] <= drawdown],
                plot_columns=[f"SPY_Rolling_Future_Return_{period_name}", f"UPRO_Rolling_Future_Return_{period_name}"],
                title=f"SPY & UPRO {period_name} Rolling Future Returns Post {drawdown} UPRO Drawdown",
                x_label="Rolling Return",
                x_tick_spacing="Auto",
                x_tick_rotation=30,
                y_label="# Of Datapoints",
                y_tick_spacing="Auto",
                y_tick_rotation=0,
                grid=True,
                legend=True,
                export_plot=False,
                plot_file_name=None,
            )

            plot_scatter(
                df=spy_upro_extrap_future[spy_upro_extrap_future["UPRO_Drawdown"] <= drawdown],
                x_plot_column=f"SPY_Rolling_Future_Return_{period_name}",
                y_plot_columns=[f"UPRO_Rolling_Future_Return_{period_name}"],
                title=f"SPY & UPRO {period_name} Rolling Future Returns Post {drawdown} UPRO Drawdown",
                x_label="SPY Rolling Return",
                x_format="Decimal",
                x_format_decimal_places=2,
                x_tick_spacing="Auto",
                x_tick_start=None,
                x_tick_rotation=30,
                y_label="UPRO Rolling Return",
                y_format="Decimal",
                y_format_decimal_places=2,
                y_tick_spacing="Auto",
                y_tick_rotation=0,
                plot_OLS_regression_line=True,
                OLS_column=f"UPRO_Rolling_Future_Return_{period_name}",
                plot_Ridge_regression_line=False,
                Ridge_column=None,
                plot_RidgeCV_regression_line=True,
                RidgeCV_column=f"UPRO_Rolling_Future_Return_{period_name}",
                regression_constant=True,
                grid=True,
                legend=True,
                export_plot=False,
                plot_file_name=None,
            )

            # Run OLS regression with statsmodels
            model = run_regression(
                df=spy_upro_extrap_future[spy_upro_extrap_future["UPRO_Drawdown"] <= drawdown],
                x_plot_column=f"SPY_Rolling_Future_Return_{period_name}",
                y_plot_column=f"UPRO_Rolling_Future_Return_{period_name}",
                regression_model="OLS-statsmodels",
                regression_constant=True,
            )
            print(model.summary())

            # Filter by drawdown
            drawdown_filter = spy_upro_extrap_future[spy_upro_extrap_future["UPRO_Drawdown"] <= drawdown]

            # Filter by period, drop rows with missing values
            future_filter = drawdown_filter[[f"UPRO_Rolling_Future_Return_{period_name}"]].dropna()

            # Find length of future dataframe
            future_length = len(future_filter)

            # Find length of future dataframe where return is positive
            positive_future_length = len(future_filter[future_filter[f"UPRO_Rolling_Future_Return_{period_name}"] > 0])

            # Calculate percentage of future returns that are positive
            positive_future_percentage = (positive_future_length / future_length) if future_length > 0 else 0

            # Add the regression results to the rolling returns stats dataframe
            intercept = model.params.iloc[0]
            # intercept_pvalue = model.pvalues.iloc[0]   # p-value for Intercept
            slope = model.params.iloc[1]
            # slope_pvalue = model.pvalues.iloc[1]       # p-value for Slope
            r_squared = model.rsquared

            rolling_returns_slope_int = pd.DataFrame({
                "Drawdown": drawdown,
                "Period": period_name,
                "Intercept": [intercept],
                # "Intercept_PValue": [intercept_pvalue],
                "Slope": [slope],
                # "Slope_PValue": [slope_pvalue],
                "R_Squared": [r_squared],
                "Positive_Future_Percentage": [positive_future_percentage],
            })
            
            rolling_returns_drawdown_stats = pd.concat([rolling_returns_drawdown_stats, rolling_returns_slope_int])

        except:
            print(f"Not enough data points for drawdown level {drawdown} and period {period_name} to run regression.")

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 2.294e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:07   Log-Likelihood:                 29954.
No. Observations:                           6240   AIC:                        -5.990e+04
Df Residuals:                               6238   BIC:                        -5.989e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         2.179e-05   2.52e-05      0.864      0.388   -2.77e-05    7.12e-05
SPY_Rolling_Future_Return_1d     2.9763      0.002   1514.736      0.000       2.972       2.980
==============================================================================
Omnibus:                     4615.098   Durbin-Watson:                   2.629
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          2417082.827
Skew:                           2.322   Prob(JB):                         0.00
Kurtosis:                      99.306   Cond. No.                         78.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.994
Model:                                       OLS   Adj. R-squared:                  0.994
Method:                            Least Squares   F-statistic:                 9.726e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:11   Log-Likelihood:                 22952.
No. Observations:                           6239   AIC:                        -4.590e+04
Df Residuals:                               6237   BIC:                        -4.589e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002   7.77e-05     -3.053      0.002      -0.000   -8.49e-05
SPY_Rolling_Future_Return_1w     2.9733      0.003    986.197      0.000       2.967       2.979
==============================================================================
Omnibus:                     2742.577   Durbin-Watson:                   0.977
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           785868.247
Skew:                          -0.875   Prob(JB):                         0.00
Kurtosis:                      57.954   Cond. No.                         39.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.988
Model:                                       OLS   Adj. R-squared:                  0.988
Method:                            Least Squares   F-statistic:                 5.043e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:15   Log-Likelihood:                 17020.
No. Observations:                           6239   AIC:                        -3.404e+04
Df Residuals:                               6237   BIC:                        -3.402e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0014      0.000     -7.114      0.000      -0.002      -0.001
SPY_Rolling_Future_Return_1m     2.9760      0.004    710.158      0.000       2.968       2.984
==============================================================================
Omnibus:                     3365.813   Durbin-Watson:                   0.336
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           388718.880
Skew:                          -1.616   Prob(JB):                         0.00
Kurtosis:                      41.534   Cond. No.                         20.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.978
Model:                                       OLS   Adj. R-squared:                  0.978
Method:                            Least Squares   F-statistic:                 2.721e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:18   Log-Likelihood:                 11931.
No. Observations:                           6224   AIC:                        -2.386e+04
Df Residuals:                               6222   BIC:                        -2.384e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0048      0.000    -10.081      0.000      -0.006      -0.004
SPY_Rolling_Future_Return_3m     3.0332      0.006    521.652      0.000       3.022       3.045
==============================================================================
Omnibus:                     1726.920   Durbin-Watson:                   0.148
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            88015.830
Skew:                           0.522   Prob(JB):                         0.00
Kurtosis:                      21.393   Cond. No.                         12.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.961
Model:                                       OLS   Adj. R-squared:                  0.961
Method:                            Least Squares   F-statistic:                 1.538e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:21   Log-Likelihood:                 7705.0
No. Observations:                           6218   AIC:                        -1.541e+04
Df Residuals:                               6216   BIC:                        -1.539e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0015      0.001     -1.542      0.123      -0.003       0.000
SPY_Rolling_Future_Return_6m     3.0328      0.008    392.132      0.000       3.018       3.048
==============================================================================
Omnibus:                     2088.631   Durbin-Watson:                   0.070
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            20755.651
Skew:                           1.316   Prob(JB):                         0.00
Kurtosis:                      11.555   Cond. No.                         8.72
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.934
Model:                                       OLS   Adj. R-squared:                  0.934
Method:                            Least Squares   F-statistic:                 8.738e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:25   Log-Likelihood:                 3111.2
No. Observations:                           6206   AIC:                            -6218.
Df Residuals:                               6204   BIC:                            -6205.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0012      0.002     -0.579      0.563      -0.005       0.003
SPY_Rolling_Future_Return_1y     3.1963      0.011    295.607      0.000       3.175       3.218
==============================================================================
Omnibus:                     1604.179   Durbin-Watson:                   0.041
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             7674.422
Skew:                           1.169   Prob(JB):                         0.00
Kurtosis:                       7.920   Cond. No.                         5.87
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.890
Model:                                       OLS   Adj. R-squared:                  0.890
Method:                            Least Squares   F-statistic:                 4.957e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:29   Log-Likelihood:                -1470.3
No. Observations:                           6105   AIC:                             2945.
Df Residuals:                               6103   BIC:                             2958.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0189      0.005     -3.886      0.000      -0.028      -0.009
SPY_Rolling_Future_Return_2y     3.4194      0.015    222.635      0.000       3.389       3.450
==============================================================================
Omnibus:                     1172.709   Durbin-Watson:                   0.024
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2727.121
Skew:                           1.085   Prob(JB):                         0.00
Kurtosis:                       5.453   Cond. No.                         4.04
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.860
Model:                                       OLS   Adj. R-squared:                  0.860
Method:                            Least Squares   F-statistic:                 3.616e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:33   Log-Likelihood:                -4765.4
No. Observations:                           5874   AIC:                             9535.
Df Residuals:                               5872   BIC:                             9548.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1518      0.009    -16.254      0.000      -0.170      -0.133
SPY_Rolling_Future_Return_3y     4.0937      0.022    190.154      0.000       4.052       4.136
==============================================================================
Omnibus:                     1619.804   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5366.747
Skew:                           1.382   Prob(JB):                         0.00
Kurtosis:                       6.779   Cond. No.                         3.30
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.843
Model:                                       OLS   Adj. R-squared:                  0.843
Method:                            Least Squares   F-statistic:                 3.027e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:36   Log-Likelihood:                -7406.0
No. Observations:                           5622   AIC:                         1.482e+04
Df Residuals:                               5620   BIC:                         1.483e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4470      0.016    -27.363      0.000      -0.479      -0.415
SPY_Rolling_Future_Return_4y     5.1190      0.029    173.984      0.000       5.061       5.177
==============================================================================
Omnibus:                     1921.179   Durbin-Watson:                   0.013
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            10320.915
Skew:                           1.544   Prob(JB):                         0.00
Kurtosis:                       8.876   Cond. No.                         2.84
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.838
Model:                                       OLS   Adj. R-squared:                  0.838
Method:                            Least Squares   F-statistic:                 2.842e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:39   Log-Likelihood:                -9619.6
No. Observations:                           5509   AIC:                         1.924e+04
Df Residuals:                               5507   BIC:                         1.926e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.9358      0.026    -35.725      0.000      -0.987      -0.884
SPY_Rolling_Future_Return_5y     6.1972      0.037    168.571      0.000       6.125       6.269
==============================================================================
Omnibus:                     1244.774   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4372.151
Skew:                           1.109   Prob(JB):                         0.00
Kurtosis:                       6.759   Cond. No.                         2.58
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 1.841e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:43   Log-Likelihood:                 25147.
No. Observations:                           5298   AIC:                        -5.029e+04
Df Residuals:                               5296   BIC:                        -5.028e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         3.538e-05   2.89e-05      1.225      0.221   -2.12e-05     9.2e-05
SPY_Rolling_Future_Return_1d     2.9771      0.002   1356.703      0.000       2.973       2.981
==============================================================================
Omnibus:                     3687.243   Durbin-Watson:                   2.648
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1791380.798
Skew:                           2.083   Prob(JB):                         0.00
Kurtosis:                      92.987   Cond. No.                         76.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.993
Model:                                       OLS   Adj. R-squared:                  0.993
Method:                            Least Squares   F-statistic:                 7.731e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:46   Log-Likelihood:                 19180.
No. Observations:                           5298   AIC:                        -3.836e+04
Df Residuals:                               5296   BIC:                        -3.834e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002   8.93e-05     -2.254      0.024      -0.000   -2.62e-05
SPY_Rolling_Future_Return_1w     2.9708      0.003    879.244      0.000       2.964       2.977
==============================================================================
Omnibus:                     2334.649   Durbin-Watson:                   0.986
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           566105.986
Skew:                          -0.920   Prob(JB):                         0.00
Kurtosis:                      53.607   Cond. No.                         37.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.987
Model:                                       OLS   Adj. R-squared:                  0.987
Method:                            Least Squares   F-statistic:                 4.039e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:50   Log-Likelihood:                 14148.
No. Observations:                           5298   AIC:                        -2.829e+04
Df Residuals:                               5296   BIC:                        -2.828e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0014      0.000     -5.972      0.000      -0.002      -0.001
SPY_Rolling_Future_Return_1m     2.9725      0.005    635.555      0.000       2.963       2.982
==============================================================================
Omnibus:                     2849.543   Durbin-Watson:                   0.332
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           279709.297
Skew:                          -1.647   Prob(JB):                         0.00
Kurtosis:                      38.443   Cond. No.                         20.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.977
Model:                                       OLS   Adj. R-squared:                  0.977
Method:                            Least Squares   F-statistic:                 2.274e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:53   Log-Likelihood:                 9942.4
No. Observations:                           5293   AIC:                        -1.988e+04
Df Residuals:                               5291   BIC:                        -1.987e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0036      0.001     -6.761      0.000      -0.005      -0.003
SPY_Rolling_Future_Return_3m     3.0229      0.006    476.837      0.000       3.010       3.035
==============================================================================
Omnibus:                     1416.127   Durbin-Watson:                   0.159
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            70526.237
Skew:                           0.467   Prob(JB):                         0.00
Kurtosis:                      20.858   Cond. No.                         12.5
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.960
Model:                                       OLS   Adj. R-squared:                  0.960
Method:                            Least Squares   F-statistic:                 1.281e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:56   Log-Likelihood:                 6392.4
No. Observations:                           5293   AIC:                        -1.278e+04
Df Residuals:                               5291   BIC:                        -1.277e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0016      0.001      1.528      0.127      -0.000       0.004
SPY_Rolling_Future_Return_6m     3.0177      0.008    357.943      0.000       3.001       3.034
==============================================================================
Omnibus:                     1752.040   Durbin-Watson:                   0.078
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            16362.810
Skew:                           1.309   Prob(JB):                         0.00
Kurtosis:                      11.206   Cond. No.                         8.50
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.937
Model:                                       OLS   Adj. R-squared:                  0.937
Method:                            Least Squares   F-statistic:                 7.845e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:39:59   Log-Likelihood:                 2767.6
No. Observations:                           5293   AIC:                            -5531.
Df Residuals:                               5291   BIC:                            -5518.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0097      0.002      4.345      0.000       0.005       0.014
SPY_Rolling_Future_Return_1y     3.1668      0.011    280.090      0.000       3.145       3.189
==============================================================================
Omnibus:                     1742.227   Durbin-Watson:                   0.051
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            10046.362
Skew:                           1.455   Prob(JB):                         0.00
Kurtosis:                       9.090   Cond. No.                         5.78
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.895
Model:                                       OLS   Adj. R-squared:                  0.895
Method:                            Least Squares   F-statistic:                 4.480e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:01   Log-Likelihood:                -843.25
No. Observations:                           5242   AIC:                             1690.
Df Residuals:                               5240   BIC:                             1704.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0070      0.005     -1.442      0.149      -0.016       0.003
SPY_Rolling_Future_Return_2y     3.3634      0.016    211.654      0.000       3.332       3.395
==============================================================================
Omnibus:                     1355.004   Durbin-Watson:                   0.031
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4718.217
Skew:                           1.274   Prob(JB):                         0.00
Kurtosis:                       6.887   Cond. No.                         4.18
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.886
Model:                                       OLS   Adj. R-squared:                  0.886
Method:                            Least Squares   F-statistic:                 3.921e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:04   Log-Likelihood:                -2559.6
No. Observations:                           5063   AIC:                             5123.
Df Residuals:                               5061   BIC:                             5136.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1032      0.008    -13.699      0.000      -0.118      -0.088
SPY_Rolling_Future_Return_3y     3.7804      0.019    198.011      0.000       3.743       3.818
==============================================================================
Omnibus:                     1581.235   Durbin-Watson:                   0.020
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8945.329
Skew:                           1.376   Prob(JB):                         0.00
Kurtosis:                       8.902   Cond. No.                         3.64
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.856
Model:                                       OLS   Adj. R-squared:                  0.856
Method:                            Least Squares   F-statistic:                 2.851e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:07   Log-Likelihood:                -4919.0
No. Observations:                           4811   AIC:                             9842.
Df Residuals:                               4809   BIC:                             9855.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3001      0.014    -22.221      0.000      -0.327      -0.274
SPY_Rolling_Future_Return_4y     4.5633      0.027    168.861      0.000       4.510       4.616
==============================================================================
Omnibus:                     2903.933   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            67539.845
Skew:                           2.452   Prob(JB):                         0.00
Kurtosis:                      20.688   Cond. No.                         3.17
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.842
Model:                                       OLS   Adj. R-squared:                  0.842
Method:                            Least Squares   F-statistic:                 2.529e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:09   Log-Likelihood:                -6916.3
No. Observations:                           4736   AIC:                         1.384e+04
Df Residuals:                               4734   BIC:                         1.385e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.6491      0.022    -29.489      0.000      -0.692      -0.606
SPY_Rolling_Future_Return_5y     5.4200      0.034    159.030      0.000       5.353       5.487
==============================================================================
Omnibus:                     2512.083   Durbin-Watson:                   0.026
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            40947.865
Skew:                           2.157   Prob(JB):                         0.00
Kurtosis:                      16.744   Cond. No.                         2.84
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 1.595e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:12   Log-Likelihood:                 22509.
No. Observations:                           4774   AIC:                        -4.501e+04
Df Residuals:                               4772   BIC:                        -4.500e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         4.703e-05   3.14e-05      1.498      0.134   -1.45e-05       0.000
SPY_Rolling_Future_Return_1d     2.9768      0.002   1263.069      0.000       2.972       2.981
==============================================================================
Omnibus:                     3250.937   Durbin-Watson:                   2.668
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1511103.280
Skew:                           2.005   Prob(JB):                         0.00
Kurtosis:                      90.067   Cond. No.                         75.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.993
Model:                                       OLS   Adj. R-squared:                  0.993
Method:                            Least Squares   F-statistic:                 6.655e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:14   Log-Likelihood:                 17170.
No. Observations:                           4774   AIC:                        -3.434e+04
Df Residuals:                               4772   BIC:                        -3.432e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002   9.63e-05     -1.928      0.054      -0.000    3.14e-06
SPY_Rolling_Future_Return_1w     2.9715      0.004    815.809      0.000       2.964       2.979
==============================================================================
Omnibus:                     1997.445   Durbin-Watson:                   0.986
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           483327.377
Skew:                          -0.802   Prob(JB):                         0.00
Kurtosis:                      52.267   Cond. No.                         37.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.987
Model:                                       OLS   Adj. R-squared:                  0.987
Method:                            Least Squares   F-statistic:                 3.515e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:17   Log-Likelihood:                 12680.
No. Observations:                           4774   AIC:                        -2.536e+04
Df Residuals:                               4772   BIC:                        -2.534e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0011      0.000     -4.425      0.000      -0.002      -0.001
SPY_Rolling_Future_Return_1m     2.9621      0.005    592.900      0.000       2.952       2.972
==============================================================================
Omnibus:                     2677.078   Durbin-Watson:                   0.336
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           262784.361
Skew:                          -1.764   Prob(JB):                         0.00
Kurtosis:                      39.175   Cond. No.                         20.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.978
Model:                                       OLS   Adj. R-squared:                  0.978
Method:                            Least Squares   F-statistic:                 2.123e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:19   Log-Likelihood:                 8982.2
No. Observations:                           4774   AIC:                        -1.796e+04
Df Residuals:                               4772   BIC:                        -1.795e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0024      0.001     -4.436      0.000      -0.004      -0.001
SPY_Rolling_Future_Return_3m     3.0170      0.007    460.797      0.000       3.004       3.030
==============================================================================
Omnibus:                     1428.917   Durbin-Watson:                   0.169
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            68289.215
Skew:                           0.660   Prob(JB):                         0.00
Kurtosis:                      21.481   Cond. No.                         12.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.960
Model:                                       OLS   Adj. R-squared:                  0.960
Method:                            Least Squares   F-statistic:                 1.138e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:23   Log-Likelihood:                 5697.1
No. Observations:                           4774   AIC:                        -1.139e+04
Df Residuals:                               4772   BIC:                        -1.138e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0029      0.001      2.545      0.011       0.001       0.005
SPY_Rolling_Future_Return_6m     3.0134      0.009    337.351      0.000       2.996       3.031
==============================================================================
Omnibus:                     1652.314   Durbin-Watson:                   0.081
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            14596.667
Skew:                           1.398   Prob(JB):                         0.00
Kurtosis:                      11.097   Cond. No.                         8.43
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.937
Model:                                       OLS   Adj. R-squared:                  0.937
Method:                            Least Squares   F-statistic:                 7.138e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:25   Log-Likelihood:                 2503.5
No. Observations:                           4774   AIC:                            -5003.
Df Residuals:                               4772   BIC:                            -4990.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0150      0.002      6.469      0.000       0.010       0.019
SPY_Rolling_Future_Return_1y     3.1483      0.012    267.163      0.000       3.125       3.171
==============================================================================
Omnibus:                     1679.284   Durbin-Watson:                   0.054
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            10737.738
Skew:                           1.532   Prob(JB):                         0.00
Kurtosis:                       9.678   Cond. No.                         5.73
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.901
Model:                                       OLS   Adj. R-squared:                  0.901
Method:                            Least Squares   F-statistic:                 4.347e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:28   Log-Likelihood:                -568.45
No. Observations:                           4753   AIC:                             1141.
Df Residuals:                               4751   BIC:                             1154.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0020      0.005      0.410      0.682      -0.008       0.012
SPY_Rolling_Future_Return_2y     3.3735      0.016    208.485      0.000       3.342       3.405
==============================================================================
Omnibus:                     1332.933   Durbin-Watson:                   0.031
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5113.538
Skew:                           1.349   Prob(JB):                         0.00
Kurtosis:                       7.306   Cond. No.                         4.22
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.904
Model:                                       OLS   Adj. R-squared:                  0.904
Method:                            Least Squares   F-statistic:                 4.346e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:31   Log-Likelihood:                -1616.8
No. Observations:                           4610   AIC:                             3238.
Df Residuals:                               4608   BIC:                             3250.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0838      0.007    -12.216      0.000      -0.097      -0.070
SPY_Rolling_Future_Return_3y     3.7321      0.018    208.472      0.000       3.697       3.767
==============================================================================
Omnibus:                      743.503   Durbin-Watson:                   0.020
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1945.772
Skew:                           0.882   Prob(JB):                         0.00
Kurtosis:                       5.649   Cond. No.                         3.79
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.911
Model:                                       OLS   Adj. R-squared:                  0.911
Method:                            Least Squares   F-statistic:                 4.442e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:34   Log-Likelihood:                -2774.5
No. Observations:                           4358   AIC:                             5553.
Df Residuals:                               4356   BIC:                             5566.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2110      0.010    -21.722      0.000      -0.230      -0.192
SPY_Rolling_Future_Return_4y     4.3324      0.021    210.772      0.000       4.292       4.373
==============================================================================
Omnibus:                      124.071   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              252.668
Skew:                          -0.181   Prob(JB):                     1.36e-55
Kurtosis:                       4.123   Cond. No.                         3.33
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.887
Model:                                       OLS   Adj. R-squared:                  0.887
Method:                            Least Squares   F-statistic:                 3.391e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:36   Log-Likelihood:                -4969.9
No. Observations:                           4317   AIC:                             9944.
Df Residuals:                               4315   BIC:                             9956.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4824      0.017    -28.175      0.000      -0.516      -0.449
SPY_Rolling_Future_Return_5y     5.0841      0.028    184.148      0.000       5.030       5.138
==============================================================================
Omnibus:                      712.684   Durbin-Watson:                   0.017
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             3462.070
Skew:                           0.712   Prob(JB):                         0.00
Kurtosis:                       7.149   Cond. No.                         2.94
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 1.487e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:38   Log-Likelihood:                 21081.
No. Observations:                           4464   AIC:                        -4.216e+04
Df Residuals:                               4462   BIC:                        -4.214e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         4.399e-05   3.22e-05      1.365      0.172   -1.92e-05       0.000
SPY_Rolling_Future_Return_1d     2.9792      0.002   1219.623      0.000       2.974       2.984
==============================================================================
Omnibus:                     2686.311   Durbin-Watson:                   2.619
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1502847.576
Skew:                           1.531   Prob(JB):                         0.00
Kurtosis:                      92.836   Cond. No.                         75.8
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.993
Model:                                       OLS   Adj. R-squared:                  0.993
Method:                            Least Squares   F-statistic:                 6.179e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:41   Log-Likelihood:                 16080.
No. Observations:                           4464   AIC:                        -3.216e+04
Df Residuals:                               4462   BIC:                        -3.214e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002    9.9e-05     -1.777      0.076      -0.000    1.82e-05
SPY_Rolling_Future_Return_1w     2.9717      0.004    786.083      0.000       2.964       2.979
==============================================================================
Omnibus:                     1977.890   Durbin-Watson:                   0.971
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           519496.172
Skew:                          -0.906   Prob(JB):                         0.00
Kurtosis:                      55.818   Cond. No.                         38.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.986
Model:                                       OLS   Adj. R-squared:                  0.986
Method:                            Least Squares   F-statistic:                 3.225e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:44   Log-Likelihood:                 11851.
No. Observations:                           4464   AIC:                        -2.370e+04
Df Residuals:                               4462   BIC:                        -2.368e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0009      0.000     -3.421      0.001      -0.001      -0.000
SPY_Rolling_Future_Return_1m     2.9520      0.005    567.903      0.000       2.942       2.962
==============================================================================
Omnibus:                     2550.136   Durbin-Watson:                   0.358
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           256159.655
Skew:                          -1.810   Prob(JB):                         0.00
Kurtosis:                      39.934   Cond. No.                         20.4
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.978
Model:                                       OLS   Adj. R-squared:                  0.978
Method:                            Least Squares   F-statistic:                 2.007e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:47   Log-Likelihood:                 8451.6
No. Observations:                           4464   AIC:                        -1.690e+04
Df Residuals:                               4462   BIC:                        -1.689e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0018      0.001     -3.146      0.002      -0.003      -0.001
SPY_Rolling_Future_Return_3m     3.0074      0.007    448.023      0.000       2.994       3.021
==============================================================================
Omnibus:                     1648.709   Durbin-Watson:                   0.182
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            57091.218
Skew:                           1.101   Prob(JB):                         0.00
Kurtosis:                      20.381   Cond. No.                         12.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.959
Model:                                       OLS   Adj. R-squared:                  0.959
Method:                            Least Squares   F-statistic:                 1.039e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:50   Log-Likelihood:                 5314.5
No. Observations:                           4464   AIC:                        -1.063e+04
Df Residuals:                               4462   BIC:                        -1.061e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0040      0.001      3.494      0.000       0.002       0.006
SPY_Rolling_Future_Return_6m     2.9973      0.009    322.291      0.000       2.979       3.016
==============================================================================
Omnibus:                     1611.744   Durbin-Watson:                   0.081
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            13013.675
Skew:                           1.500   Prob(JB):                         0.00
Kurtosis:                      10.808   Cond. No.                         8.46
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.936
Model:                                       OLS   Adj. R-squared:                  0.936
Method:                            Least Squares   F-statistic:                 6.531e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:55   Log-Likelihood:                 2342.5
No. Observations:                           4464   AIC:                            -4681.
Df Residuals:                               4462   BIC:                            -4668.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0162      0.002      6.870      0.000       0.012       0.021
SPY_Rolling_Future_Return_1y     3.1343      0.012    255.552      0.000       3.110       3.158
==============================================================================
Omnibus:                     1688.563   Durbin-Watson:                   0.054
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            11825.259
Skew:                           1.634   Prob(JB):                         0.00
Kurtosis:                      10.273   Cond. No.                         5.76
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.907
Model:                                       OLS   Adj. R-squared:                  0.907
Method:                            Least Squares   F-statistic:                 4.354e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:40:58   Log-Likelihood:                -457.03
No. Observations:                           4456   AIC:                             918.1
Df Residuals:                               4454   BIC:                             930.9
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0011      0.005      0.223      0.824      -0.009       0.011
SPY_Rolling_Future_Return_2y     3.4516      0.017    208.651      0.000       3.419       3.484
==============================================================================
Omnibus:                     1243.382   Durbin-Watson:                   0.033
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4602.362
Skew:                           1.354   Prob(JB):                         0.00
Kurtosis:                       7.178   Cond. No.                         4.25
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.907
Model:                                       OLS   Adj. R-squared:                  0.907
Method:                            Least Squares   F-statistic:                 4.275e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:01   Log-Likelihood:                -1484.9
No. Observations:                           4388   AIC:                             2974.
Df Residuals:                               4386   BIC:                             2987.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0806      0.007    -11.574      0.000      -0.094      -0.067
SPY_Rolling_Future_Return_3y     3.7638      0.018    206.771      0.000       3.728       3.799
==============================================================================
Omnibus:                      756.129   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1962.708
Skew:                           0.940   Prob(JB):                         0.00
Kurtosis:                       5.683   Cond. No.                         3.81
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.923
Model:                                       OLS   Adj. R-squared:                  0.923
Method:                            Least Squares   F-statistic:                 4.998e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:04   Log-Likelihood:                -2359.9
No. Observations:                           4145   AIC:                             4724.
Df Residuals:                               4143   BIC:                             4736.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2199      0.009    -23.590      0.000      -0.238      -0.202
SPY_Rolling_Future_Return_4y     4.4469      0.020    223.555      0.000       4.408       4.486
==============================================================================
Omnibus:                      131.101   Durbin-Watson:                   0.024
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              321.129
Skew:                          -0.122   Prob(JB):                     1.85e-70
Kurtosis:                       4.342   Cond. No.                         3.35
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.896
Model:                                       OLS   Adj. R-squared:                  0.896
Method:                            Least Squares   F-statistic:                 3.555e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:06   Log-Likelihood:                -4610.5
No. Observations:                           4126   AIC:                             9225.
Df Residuals:                               4124   BIC:                             9238.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5058      0.017    -29.865      0.000      -0.539      -0.473
SPY_Rolling_Future_Return_5y     5.2121      0.028    188.535      0.000       5.158       5.266
==============================================================================
Omnibus:                      688.469   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             3305.383
Skew:                           0.724   Prob(JB):                         0.00
Kurtosis:                       7.139   Cond. No.                         2.96
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 1.337e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:09   Log-Likelihood:                 18280.
No. Observations:                           3870   AIC:                        -3.656e+04
Df Residuals:                               3868   BIC:                        -3.654e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         5.793e-05   3.46e-05      1.675      0.094   -9.86e-06       0.000
SPY_Rolling_Future_Return_1d     2.9842      0.003   1156.432      0.000       2.979       2.989
==============================================================================
Omnibus:                     2832.298   Durbin-Watson:                   2.689
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1197723.159
Skew:                           2.308   Prob(JB):                         0.00
Kurtosis:                      89.061   Cond. No.                         74.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.993
Model:                                       OLS   Adj. R-squared:                  0.993
Method:                            Least Squares   F-statistic:                 5.122e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:11   Log-Likelihood:                 13824.
No. Observations:                           3870   AIC:                        -2.764e+04
Df Residuals:                               3868   BIC:                        -2.763e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0001      0.000     -0.995      0.320      -0.000       0.000
SPY_Rolling_Future_Return_1w     2.9703      0.004    715.673      0.000       2.962       2.978
==============================================================================
Omnibus:                     1719.279   Durbin-Watson:                   1.003
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           452041.648
Skew:                          -0.906   Prob(JB):                         0.00
Kurtosis:                      55.916   Cond. No.                         38.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.986
Model:                                       OLS   Adj. R-squared:                  0.986
Method:                            Least Squares   F-statistic:                 2.738e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:14   Log-Likelihood:                 10216.
No. Observations:                           3870   AIC:                        -2.043e+04
Df Residuals:                               3868   BIC:                        -2.042e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008      0.000     -2.723      0.006      -0.001      -0.000
SPY_Rolling_Future_Return_1m     2.9438      0.006    523.248      0.000       2.933       2.955
==============================================================================
Omnibus:                     2106.264   Durbin-Watson:                   0.385
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           199242.609
Skew:                          -1.680   Prob(JB):                         0.00
Kurtosis:                      37.990   Cond. No.                         20.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.978
Model:                                       OLS   Adj. R-squared:                  0.978
Method:                            Least Squares   F-statistic:                 1.701e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:17   Log-Likelihood:                 7304.4
No. Observations:                           3870   AIC:                        -1.460e+04
Df Residuals:                               3868   BIC:                        -1.459e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0012      0.001     -1.903      0.057      -0.002    3.47e-05
SPY_Rolling_Future_Return_3m     2.9839      0.007    412.396      0.000       2.970       2.998
==============================================================================
Omnibus:                     1374.843   Durbin-Watson:                   0.193
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            43043.094
Skew:                           1.059   Prob(JB):                         0.00
Kurtosis:                      19.200   Cond. No.                         12.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.957
Model:                                       OLS   Adj. R-squared:                  0.957
Method:                            Least Squares   F-statistic:                 8.588e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:20   Log-Likelihood:                 4530.7
No. Observations:                           3870   AIC:                            -9057.
Df Residuals:                               3868   BIC:                            -9045.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0067      0.001      5.298      0.000       0.004       0.009
SPY_Rolling_Future_Return_6m     2.9569      0.010    293.052      0.000       2.937       2.977
==============================================================================
Omnibus:                     1292.264   Durbin-Watson:                   0.081
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             9388.095
Skew:                           1.396   Prob(JB):                         0.00
Kurtosis:                      10.101   Cond. No.                         8.37
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.933
Model:                                       OLS   Adj. R-squared:                  0.933
Method:                            Least Squares   F-statistic:                 5.407e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:23   Log-Likelihood:                 1957.2
No. Observations:                           3870   AIC:                            -3910.
Df Residuals:                               3868   BIC:                            -3898.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0175      0.003      6.790      0.000       0.012       0.023
SPY_Rolling_Future_Return_1y     3.1290      0.013    232.519      0.000       3.103       3.155
==============================================================================
Omnibus:                     1514.284   Durbin-Watson:                   0.057
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            11128.001
Skew:                           1.682   Prob(JB):                         0.00
Kurtosis:                      10.596   Cond. No.                         5.77
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.912
Model:                                       OLS   Adj. R-squared:                  0.912
Method:                            Least Squares   F-statistic:                 4.021e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:26   Log-Likelihood:                -377.88
No. Observations:                           3870   AIC:                             759.8
Df Residuals:                               3868   BIC:                             772.3
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0141      0.005     -2.680      0.007      -0.024      -0.004
SPY_Rolling_Future_Return_2y     3.5920      0.018    200.520      0.000       3.557       3.627
==============================================================================
Omnibus:                     1083.656   Durbin-Watson:                   0.036
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             3627.502
Skew:                           1.394   Prob(JB):                         0.00
Kurtosis:                       6.837   Cond. No.                         4.30
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.910
Model:                                       OLS   Adj. R-squared:                  0.910
Method:                            Least Squares   F-statistic:                 3.905e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:28   Log-Likelihood:                -1326.0
No. Observations:                           3864   AIC:                             2656.
Df Residuals:                               3862   BIC:                             2668.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0866      0.007    -11.721      0.000      -0.101      -0.072
SPY_Rolling_Future_Return_3y     3.8539      0.020    197.605      0.000       3.816       3.892
==============================================================================
Omnibus:                      647.380   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1443.977
Skew:                           0.967   Prob(JB):                         0.00
Kurtosis:                       5.286   Cond. No.                         3.80
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.940
Model:                                       OLS   Adj. R-squared:                  0.940
Method:                            Least Squares   F-statistic:                 5.809e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:32   Log-Likelihood:                -1763.9
No. Observations:                           3695   AIC:                             3532.
Df Residuals:                               3693   BIC:                             3544.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2198      0.009    -24.580      0.000      -0.237      -0.202
SPY_Rolling_Future_Return_4y     4.5977      0.019    241.020      0.000       4.560       4.635
==============================================================================
Omnibus:                      140.904   Durbin-Watson:                   0.030
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              414.925
Skew:                           0.068   Prob(JB):                     7.95e-91
Kurtosis:                       4.636   Cond. No.                         3.32
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.907
Model:                                       OLS   Adj. R-squared:                  0.907
Method:                            Least Squares   F-statistic:                 3.594e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:36   Log-Likelihood:                -3995.9
No. Observations:                           3695   AIC:                             7996.
Df Residuals:                               3693   BIC:                             8008.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5138      0.017    -30.033      0.000      -0.547      -0.480
SPY_Rolling_Future_Return_5y     5.3991      0.028    189.569      0.000       5.343       5.455
==============================================================================
Omnibus:                      550.522   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2876.716
Skew:                           0.608   Prob(JB):                         0.00
Kurtosis:                       7.148   Cond. No.                         2.96
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 1.059e+06
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:39   Log-Likelihood:                 14432.
No. Observations:                           3070   AIC:                        -2.886e+04
Df Residuals:                               3068   BIC:                        -2.885e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         7.423e-05   3.97e-05      1.869      0.062   -3.66e-06       0.000
SPY_Rolling_Future_Return_1d     2.9824      0.003   1028.892      0.000       2.977       2.988
==============================================================================
Omnibus:                     2273.338   Durbin-Watson:                   2.537
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          1052903.376
Skew:                           2.319   Prob(JB):                         0.00
Kurtosis:                      93.607   Cond. No.                         73.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.992
Model:                                       OLS   Adj. R-squared:                  0.992
Method:                            Least Squares   F-statistic:                 3.762e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:41   Log-Likelihood:                 10750.
No. Observations:                           3070   AIC:                        -2.150e+04
Df Residuals:                               3068   BIC:                        -2.148e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0001      0.000     -1.042      0.297      -0.000       0.000
SPY_Rolling_Future_Return_1w     2.9677      0.005    613.327      0.000       2.958       2.977
==============================================================================
Omnibus:                     1399.668   Durbin-Watson:                   1.060
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           313387.446
Skew:                          -0.996   Prob(JB):                         0.00
Kurtosis:                      52.457   Cond. No.                         36.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.986
Model:                                       OLS   Adj. R-squared:                  0.986
Method:                            Least Squares   F-statistic:                 2.086e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:44   Log-Likelihood:                 7938.4
No. Observations:                           3070   AIC:                        -1.587e+04
Df Residuals:                               3068   BIC:                        -1.586e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.000     -1.709      0.088      -0.001    8.37e-05
SPY_Rolling_Future_Return_1m     2.9396      0.006    456.780      0.000       2.927       2.952
==============================================================================
Omnibus:                     1387.377   Durbin-Watson:                   0.407
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           109399.191
Skew:                          -1.256   Prob(JB):                         0.00
Kurtosis:                      32.136   Cond. No.                         19.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.977
Model:                                       OLS   Adj. R-squared:                  0.977
Method:                            Least Squares   F-statistic:                 1.313e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:46   Log-Likelihood:                 5610.6
No. Observations:                           3070   AIC:                        -1.122e+04
Df Residuals:                               3068   BIC:                        -1.121e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0004      0.001      0.591      0.554      -0.001       0.002
SPY_Rolling_Future_Return_3m     2.9878      0.008    362.351      0.000       2.972       3.004
==============================================================================
Omnibus:                     1129.279   Durbin-Watson:                   0.193
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            29411.736
Skew:                           1.166   Prob(JB):                         0.00
Kurtosis:                      17.983   Cond. No.                         11.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.956
Model:                                       OLS   Adj. R-squared:                  0.956
Method:                            Least Squares   F-statistic:                 6.658e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:49   Log-Likelihood:                 3373.7
No. Observations:                           3070   AIC:                            -6743.
Df Residuals:                               3068   BIC:                            -6731.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0138      0.002      9.064      0.000       0.011       0.017
SPY_Rolling_Future_Return_6m     2.9411      0.011    258.025      0.000       2.919       2.963
==============================================================================
Omnibus:                      862.192   Durbin-Watson:                   0.075
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5063.517
Skew:                           1.201   Prob(JB):                         0.00
Kurtosis:                       8.815   Cond. No.                         7.84
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.932
Model:                                       OLS   Adj. R-squared:                  0.932
Method:                            Least Squares   F-statistic:                 4.197e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:52   Log-Likelihood:                 1581.3
No. Observations:                           3070   AIC:                            -3159.
Df Residuals:                               3068   BIC:                            -3146.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0132      0.003      4.454      0.000       0.007       0.019
SPY_Rolling_Future_Return_1y     3.1761      0.016    204.863      0.000       3.146       3.207
==============================================================================
Omnibus:                     1211.028   Durbin-Watson:                   0.065
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            10512.118
Skew:                           1.634   Prob(JB):                         0.00
Kurtosis:                      11.456   Cond. No.                         5.99
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.932
Model:                                       OLS   Adj. R-squared:                  0.932
Method:                            Least Squares   F-statistic:                 4.210e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:54   Log-Likelihood:                 230.84
No. Observations:                           3070   AIC:                            -457.7
Df Residuals:                               3068   BIC:                            -445.6
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1524      0.006    -26.792      0.000      -0.164      -0.141
SPY_Rolling_Future_Return_2y     4.1801      0.020    205.186      0.000       4.140       4.220
==============================================================================
Omnibus:                      971.657   Durbin-Watson:                   0.048
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5573.550
Skew:                           1.382   Prob(JB):                         0.00
Kurtosis:                       8.994   Cond. No.                         5.23
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.906
Model:                                       OLS   Adj. R-squared:                  0.906
Method:                            Least Squares   F-statistic:                 2.942e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:41:57   Log-Likelihood:                -1092.6
No. Observations:                           3070   AIC:                             2189.
Df Residuals:                               3068   BIC:                             2201.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1351      0.009    -15.221      0.000      -0.153      -0.118
SPY_Rolling_Future_Return_3y     4.1098      0.024    171.517      0.000       4.063       4.157
==============================================================================
Omnibus:                      523.451   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              948.262
Skew:                           1.070   Prob(JB):                    1.22e-206
Kurtosis:                       4.683   Cond. No.                         4.13
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.942
Model:                                       OLS   Adj. R-squared:                  0.942
Method:                            Least Squares   F-statistic:                 4.920e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:00   Log-Likelihood:                -1478.4
No. Observations:                           3055   AIC:                             2961.
Df Residuals:                               3053   BIC:                             2973.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2228      0.010    -21.592      0.000      -0.243      -0.203
SPY_Rolling_Future_Return_4y     4.6897      0.021    221.806      0.000       4.648       4.731
==============================================================================
Omnibus:                      120.398   Durbin-Watson:                   0.033
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              311.753
Skew:                           0.172   Prob(JB):                     2.01e-68
Kurtosis:                       4.527   Cond. No.                         3.39
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.919
Model:                                       OLS   Adj. R-squared:                  0.919
Method:                            Least Squares   F-statistic:                 3.475e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:02   Log-Likelihood:                -3195.7
No. Observations:                           3055   AIC:                             6395.
Df Residuals:                               3053   BIC:                             6407.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5516      0.019    -29.727      0.000      -0.588      -0.515
SPY_Rolling_Future_Return_5y     5.6648      0.030    186.401      0.000       5.605       5.724
==============================================================================
Omnibus:                      454.885   Durbin-Watson:                   0.025
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2119.392
Skew:                           0.640   Prob(JB):                         0.00
Kurtosis:                       6.875   Cond. No.                         3.02
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 7.538e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:05   Log-Likelihood:                 10930.
No. Observations:                           2365   AIC:                        -2.186e+04
Df Residuals:                               2363   BIC:                        -2.184e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         8.911e-05    4.9e-05      1.819      0.069   -6.98e-06       0.000
SPY_Rolling_Future_Return_1d     2.9778      0.003    868.207      0.000       2.971       2.985
==============================================================================
Omnibus:                     1557.931   Durbin-Watson:                   2.718
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           638187.039
Skew:                           1.869   Prob(JB):                         0.00
Kurtosis:                      83.389   Cond. No.                         70.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.991
Model:                                       OLS   Adj. R-squared:                  0.991
Method:                            Least Squares   F-statistic:                 2.714e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:09   Log-Likelihood:                 8089.3
No. Observations:                           2365   AIC:                        -1.617e+04
Df Residuals:                               2363   BIC:                        -1.616e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         -7.27e-05      0.000     -0.446      0.656      -0.000       0.000
SPY_Rolling_Future_Return_1w     2.9631      0.006    520.948      0.000       2.952       2.974
==============================================================================
Omnibus:                      893.210   Durbin-Watson:                   1.046
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           167515.158
Skew:                          -0.620   Prob(JB):                         0.00
Kurtosis:                      44.212   Cond. No.                         34.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.984
Model:                                       OLS   Adj. R-squared:                  0.984
Method:                            Least Squares   F-statistic:                 1.475e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:12   Log-Likelihood:                 5932.2
No. Observations:                           2365   AIC:                        -1.186e+04
Df Residuals:                               2363   BIC:                        -1.185e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0007      0.000     -1.731      0.084      -0.002     9.4e-05
SPY_Rolling_Future_Return_1m     2.9413      0.008    384.059      0.000       2.926       2.956
==============================================================================
Omnibus:                      954.846   Durbin-Watson:                   0.364
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            63616.774
Skew:                          -1.056   Prob(JB):                         0.00
Kurtosis:                      28.320   Cond. No.                         18.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.976
Model:                                       OLS   Adj. R-squared:                  0.976
Method:                            Least Squares   F-statistic:                 9.454e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:16   Log-Likelihood:                 4110.8
No. Observations:                           2365   AIC:                            -8218.
Df Residuals:                               2363   BIC:                            -8206.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0012      0.001      1.376      0.169      -0.001       0.003
SPY_Rolling_Future_Return_3m     2.9865      0.010    307.476      0.000       2.967       3.006
==============================================================================
Omnibus:                      813.502   Durbin-Watson:                   0.169
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            16777.462
Skew:                           1.111   Prob(JB):                         0.00
Kurtosis:                      15.858   Cond. No.                         11.1
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.954
Model:                                       OLS   Adj. R-squared:                  0.954
Method:                            Least Squares   F-statistic:                 4.945e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:19   Log-Likelihood:                 2608.9
No. Observations:                           2365   AIC:                            -5214.
Df Residuals:                               2363   BIC:                            -5202.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0035      0.002      1.918      0.055   -7.76e-05       0.007
SPY_Rolling_Future_Return_6m     3.0747      0.014    222.382      0.000       3.048       3.102
==============================================================================
Omnibus:                      829.894   Durbin-Watson:                   0.069
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             7402.918
Skew:                           1.399   Prob(JB):                         0.00
Kurtosis:                      11.203   Cond. No.                         8.39
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.941
Model:                                       OLS   Adj. R-squared:                  0.941
Method:                            Least Squares   F-statistic:                 3.754e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:22   Log-Likelihood:                 1607.5
No. Observations:                           2365   AIC:                            -3211.
Df Residuals:                               2363   BIC:                            -3199.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0550      0.003    -16.380      0.000      -0.062      -0.048
SPY_Rolling_Future_Return_1y     3.5928      0.019    193.751      0.000       3.556       3.629
==============================================================================
Omnibus:                      724.877   Durbin-Watson:                   0.076
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            11317.638
Skew:                           1.018   Prob(JB):                         0.00
Kurtosis:                      13.522   Cond. No.                         7.46
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.947
Model:                                       OLS   Adj. R-squared:                  0.947
Method:                            Least Squares   F-statistic:                 4.183e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:25   Log-Likelihood:                 728.42
No. Observations:                           2365   AIC:                            -1453.
Df Residuals:                               2363   BIC:                            -1441.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3422      0.007    -49.481      0.000      -0.356      -0.329
SPY_Rolling_Future_Return_2y     4.7991      0.023    204.536      0.000       4.753       4.845
==============================================================================
Omnibus:                      261.162   Durbin-Watson:                   0.050
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1787.524
Skew:                          -0.271   Prob(JB):                         0.00
Kurtosis:                       7.225   Cond. No.                         6.82
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.901
Model:                                       OLS   Adj. R-squared:                  0.901
Method:                            Least Squares   F-statistic:                 2.157e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:29   Log-Likelihood:                -648.60
No. Observations:                           2365   AIC:                             1301.
Df Residuals:                               2363   BIC:                             1313.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3554      0.013    -28.106      0.000      -0.380      -0.331
SPY_Rolling_Future_Return_3y     4.7081      0.032    146.883      0.000       4.645       4.771
==============================================================================
Omnibus:                      350.092   Durbin-Watson:                   0.028
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              725.322
Skew:                           0.885   Prob(JB):                    3.15e-158
Kurtosis:                       5.057   Cond. No.                         5.47
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.925
Model:                                       OLS   Adj. R-squared:                  0.925
Method:                            Least Squares   F-statistic:                 2.909e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:32   Log-Likelihood:                -1339.2
No. Observations:                           2365   AIC:                             2682.
Df Residuals:                               2363   BIC:                             2694.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1851      0.014    -12.911      0.000      -0.213      -0.157
SPY_Rolling_Future_Return_4y     4.6424      0.027    170.563      0.000       4.589       4.696
==============================================================================
Omnibus:                       55.462   Durbin-Watson:                   0.030
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              118.230
Skew:                           0.081   Prob(JB):                     2.12e-26
Kurtosis:                       4.083   Cond. No.                         3.69
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.917
Model:                                       OLS   Adj. R-squared:                  0.917
Method:                            Least Squares   F-statistic:                 2.615e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:35   Log-Likelihood:                -2565.1
No. Observations:                           2365   AIC:                             5134.
Df Residuals:                               2363   BIC:                             5146.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4704      0.023    -20.190      0.000      -0.516      -0.425
SPY_Rolling_Future_Return_5y     5.6929      0.035    161.724      0.000       5.624       5.762
==============================================================================
Omnibus:                      310.990   Durbin-Watson:                   0.028
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1362.143
Skew:                           0.566   Prob(JB):                    1.64e-296
Kurtosis:                       6.542   Cond. No.                         3.12
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.997
Model:                                       OLS   Adj. R-squared:                  0.997
Method:                            Least Squares   F-statistic:                 4.359e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:38   Log-Likelihood:                 6602.3
No. Observations:                           1478   AIC:                        -1.320e+04
Df Residuals:                               1476   BIC:                        -1.319e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0001   7.23e-05      1.541      0.123   -3.04e-05       0.000
SPY_Rolling_Future_Return_1d     2.9735      0.005    660.249      0.000       2.965       2.982
==============================================================================
Omnibus:                      758.987   Durbin-Watson:                   2.689
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           237450.371
Skew:                           1.143   Prob(JB):                         0.00
Kurtosis:                      65.053   Cond. No.                         62.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.990
Model:                                       OLS   Adj. R-squared:                  0.990
Method:                            Least Squares   F-statistic:                 1.474e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:40   Log-Likelihood:                 4775.6
No. Observations:                           1478   AIC:                            -9547.
Df Residuals:                               1476   BIC:                            -9537.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                         4.324e-06      0.000      0.017      0.986      -0.000       0.000
SPY_Rolling_Future_Return_1w     2.9572      0.008    383.892      0.000       2.942       2.972
==============================================================================
Omnibus:                      517.459   Durbin-Watson:                   1.023
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            52535.710
Skew:                          -0.625   Prob(JB):                         0.00
Kurtosis:                      32.181   Cond. No.                         31.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.983
Model:                                       OLS   Adj. R-squared:                  0.983
Method:                            Least Squares   F-statistic:                 8.446e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:43   Log-Likelihood:                 3586.0
No. Observations:                           1478   AIC:                            -7168.
Df Residuals:                               1476   BIC:                            -7157.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0029      0.001     -5.071      0.000      -0.004      -0.002
SPY_Rolling_Future_Return_1m     3.0166      0.010    290.613      0.000       2.996       3.037
==============================================================================
Omnibus:                      950.842   Durbin-Watson:                   0.291
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            19703.166
Skew:                          -2.648   Prob(JB):                         0.00
Kurtosis:                      20.085   Cond. No.                         18.7
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.980
Model:                                       OLS   Adj. R-squared:                  0.980
Method:                            Least Squares   F-statistic:                 7.176e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:46   Log-Likelihood:                 2747.3
No. Observations:                           1478   AIC:                            -5491.
Df Residuals:                               1476   BIC:                            -5480.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0120      0.001    -11.212      0.000      -0.014      -0.010
SPY_Rolling_Future_Return_3m     3.2238      0.012    267.877      0.000       3.200       3.247
==============================================================================
Omnibus:                      396.149   Durbin-Watson:                   0.203
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             3162.485
Skew:                          -1.020   Prob(JB):                         0.00
Kurtosis:                       9.869   Cond. No.                         12.3
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.971
Model:                                       OLS   Adj. R-squared:                  0.971
Method:                            Least Squares   F-statistic:                 4.870e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:49   Log-Likelihood:                 1994.5
No. Observations:                           1478   AIC:                            -3985.
Df Residuals:                               1476   BIC:                            -3974.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0411      0.002    -19.749      0.000      -0.045      -0.037
SPY_Rolling_Future_Return_6m     3.5209      0.016    220.685      0.000       3.490       3.552
==============================================================================
Omnibus:                      302.933   Durbin-Watson:                   0.139
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1794.276
Skew:                          -0.817   Prob(JB):                         0.00
Kurtosis:                       8.144   Cond. No.                         9.83
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.945
Model:                                       OLS   Adj. R-squared:                  0.945
Method:                            Least Squares   F-statistic:                 2.529e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:52   Log-Likelihood:                 1207.7
No. Observations:                           1478   AIC:                            -2411.
Df Residuals:                               1476   BIC:                            -2401.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1638      0.005    -32.560      0.000      -0.174      -0.154
SPY_Rolling_Future_Return_1y     4.1147      0.026    159.043      0.000       4.064       4.165
==============================================================================
Omnibus:                      635.753   Durbin-Watson:                   0.055
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4012.236
Skew:                          -1.899   Prob(JB):                         0.00
Kurtosis:                      10.122   Cond. No.                         9.55
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.940
Model:                                       OLS   Adj. R-squared:                  0.940
Method:                            Least Squares   F-statistic:                 2.321e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:54   Log-Likelihood:                 416.75
No. Observations:                           1478   AIC:                            -829.5
Df Residuals:                               1476   BIC:                            -818.9
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5307      0.012    -45.546      0.000      -0.554      -0.508
SPY_Rolling_Future_Return_2y     5.2902      0.035    152.337      0.000       5.222       5.358
==============================================================================
Omnibus:                      349.925   Durbin-Watson:                   0.045
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              914.237
Skew:                          -1.242   Prob(JB):                    2.99e-199
Kurtosis:                       5.945   Cond. No.                         8.01
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.884
Model:                                       OLS   Adj. R-squared:                  0.884
Method:                            Least Squares   F-statistic:                 1.127e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:56   Log-Likelihood:                -267.98
No. Observations:                           1478   AIC:                             540.0
Df Residuals:                               1476   BIC:                             550.6
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.0261      0.027    -38.067      0.000      -1.079      -0.973
SPY_Rolling_Future_Return_3y     6.1824      0.058    106.180      0.000       6.068       6.297
==============================================================================
Omnibus:                       75.106   Durbin-Watson:                   0.032
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               91.663
Skew:                          -0.514   Prob(JB):                     1.25e-20
Kurtosis:                       3.657   Cond. No.                         9.26
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.877
Model:                                       OLS   Adj. R-squared:                  0.877
Method:                            Least Squares   F-statistic:                 1.049e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:42:59   Log-Likelihood:                -459.80
No. Observations:                           1478   AIC:                             923.6
Df Residuals:                               1476   BIC:                             934.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3239      0.040    -33.279      0.000      -1.402      -1.246
SPY_Rolling_Future_Return_4y     6.5227      0.064    102.411      0.000       6.398       6.648
==============================================================================
Omnibus:                      316.558   Durbin-Watson:                   0.032
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              745.315
Skew:                          -1.168   Prob(JB):                    1.43e-162
Kurtosis:                       5.578   Cond. No.                         10.2
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.895
Model:                                       OLS   Adj. R-squared:                  0.895
Method:                            Least Squares   F-statistic:                 1.257e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:01   Log-Likelihood:                -1477.7
No. Observations:                           1478   AIC:                             2959.
Df Residuals:                               1476   BIC:                             2970.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3148      0.048    -27.521      0.000      -1.409      -1.221
SPY_Rolling_Future_Return_5y     6.8269      0.061    112.133      0.000       6.707       6.946
==============================================================================
Omnibus:                      180.874   Durbin-Watson:                   0.042
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              812.227
Skew:                           0.496   Prob(JB):                    4.24e-177
Kurtosis:                       6.493   Cond. No.                         5.57
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1d   R-squared:                       0.998
Model:                                       OLS   Adj. R-squared:                  0.998
Method:                            Least Squares   F-statistic:                 2.300e+05
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:03   Log-Likelihood:                 2171.6
No. Observations:                            492   AIC:                            -4339.
Df Residuals:                                490   BIC:                            -4331.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0001      0.000      0.853      0.394      -0.000       0.000
SPY_Rolling_Future_Return_1d     2.9737      0.006    479.627      0.000       2.962       2.986
==============================================================================
Omnibus:                      207.234   Durbin-Watson:                   2.771
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            96838.420
Skew:                           0.200   Prob(JB):                         0.00
Kurtosis:                      71.729   Cond. No.                         46.8
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1w   R-squared:                       0.990
Model:                                       OLS   Adj. R-squared:                  0.990
Method:                            Least Squares   F-statistic:                 4.756e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:06   Log-Likelihood:                 1471.4
No. Observations:                            492   AIC:                            -2939.
Df Residuals:                                490   BIC:                            -2930.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0009      0.001     -1.610      0.108      -0.002       0.000
SPY_Rolling_Future_Return_1w     2.9981      0.014    218.085      0.000       2.971       3.025
==============================================================================
Omnibus:                      246.103   Durbin-Watson:                   1.337
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             7382.619
Skew:                          -1.556   Prob(JB):                         0.00
Kurtosis:                      21.720   Cond. No.                         25.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1m   R-squared:                       0.977
Model:                                       OLS   Adj. R-squared:                  0.977
Method:                            Least Squares   F-statistic:                 2.054e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:08   Log-Likelihood:                 1034.0
No. Observations:                            492   AIC:                            -2064.
Df Residuals:                                490   BIC:                            -2056.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0072      0.001     -5.051      0.000      -0.010      -0.004
SPY_Rolling_Future_Return_1m     3.0527      0.021    143.318      0.000       3.011       3.095
==============================================================================
Omnibus:                      246.880   Durbin-Watson:                   0.321
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             2046.076
Skew:                          -2.018   Prob(JB):                         0.00
Kurtosis:                      12.139   Cond. No.                         15.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3m   R-squared:                       0.976
Model:                                       OLS   Adj. R-squared:                  0.976
Method:                            Least Squares   F-statistic:                 2.001e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:10   Log-Likelihood:                 794.88
No. Observations:                            492   AIC:                            -1586.
Df Residuals:                                490   BIC:                            -1577.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0291      0.003    -11.077      0.000      -0.034      -0.024
SPY_Rolling_Future_Return_3m     3.3540      0.024    141.466      0.000       3.307       3.401
==============================================================================
Omnibus:                       75.895   Durbin-Watson:                   0.347
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              485.942
Skew:                          -0.458   Prob(JB):                    3.01e-106
Kurtosis:                       7.782   Cond. No.                         11.0
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_6m   R-squared:                       0.968
Model:                                       OLS   Adj. R-squared:                  0.968
Method:                            Least Squares   F-statistic:                 1.482e+04
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):               0.00
Time:                                   14:43:13   Log-Likelihood:                 557.26
No. Observations:                            492   AIC:                            -1111.
Df Residuals:                                490   BIC:                            -1102.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0991      0.006    -17.983      0.000      -0.110      -0.088
SPY_Rolling_Future_Return_6m     3.8451      0.032    121.754      0.000       3.783       3.907
==============================================================================
Omnibus:                       89.885   Durbin-Watson:                   0.146
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              142.024
Skew:                          -1.149   Prob(JB):                     1.45e-31
Kurtosis:                       4.284   Cond. No.                         9.13
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_1y   R-squared:                       0.926
Model:                                       OLS   Adj. R-squared:                  0.926
Method:                            Least Squares   F-statistic:                     6156.
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):          1.39e-279
Time:                                   14:43:15   Log-Likelihood:                 241.59
No. Observations:                            492   AIC:                            -479.2
Df Residuals:                                490   BIC:                            -470.8
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2182      0.013    -16.797      0.000      -0.244      -0.193
SPY_Rolling_Future_Return_1y     4.1968      0.053     78.460      0.000       4.092       4.302
==============================================================================
Omnibus:                      129.416   Durbin-Watson:                   0.094
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              301.681
Skew:                          -1.350   Prob(JB):                     3.10e-66
Kurtosis:                       5.725   Cond. No.                         8.35
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_2y   R-squared:                       0.943
Model:                                       OLS   Adj. R-squared:                  0.943
Method:                            Least Squares   F-statistic:                     8092.
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):          8.78e-307
Time:                                   14:43:18   Log-Likelihood:                 42.960
No. Observations:                            492   AIC:                            -81.92
Df Residuals:                                490   BIC:                            -73.52
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5693      0.022    -26.067      0.000      -0.612      -0.526
SPY_Rolling_Future_Return_2y     5.1161      0.057     89.953      0.000       5.004       5.228
==============================================================================
Omnibus:                       62.745   Durbin-Watson:                   0.075
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              117.512
Skew:                          -0.751   Prob(JB):                     3.04e-26
Kurtosis:                       4.864   Cond. No.                         6.36
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_3y   R-squared:                       0.895
Model:                                       OLS   Adj. R-squared:                  0.895
Method:                            Least Squares   F-statistic:                     4166.
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):          1.06e-241
Time:                                   14:43:21   Log-Likelihood:                -134.24
No. Observations:                            492   AIC:                             272.5
Df Residuals:                                490   BIC:                             280.9
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.2645      0.049    -25.786      0.000      -1.361      -1.168
SPY_Rolling_Future_Return_3y     6.2301      0.097     64.542      0.000       6.040       6.420
==============================================================================
Omnibus:                       10.395   Durbin-Watson:                   0.052
Prob(Omnibus):                  0.006   Jarque-Bera (JB):               14.422
Skew:                           0.184   Prob(JB):                     0.000738
Kurtosis:                       3.754   Cond. No.                         8.34
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_4y   R-squared:                       0.874
Model:                                       OLS   Adj. R-squared:                  0.874
Method:                            Least Squares   F-statistic:                     3409.
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):          8.08e-223
Time:                                   14:43:23   Log-Likelihood:                -226.89
No. Observations:                            492   AIC:                             457.8
Df Residuals:                                490   BIC:                             466.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -2.5519      0.101    -25.150      0.000      -2.751      -2.353
SPY_Rolling_Future_Return_4y     7.9918      0.137     58.384      0.000       7.723       8.261
==============================================================================
Omnibus:                       35.379   Durbin-Watson:                   0.062
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               41.084
Skew:                          -0.696   Prob(JB):                     1.20e-09
Kurtosis:                       3.261   Cond. No.                         12.2
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

png

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     UPRO_Rolling_Future_Return_5y   R-squared:                       0.871
Model:                                       OLS   Adj. R-squared:                  0.870
Method:                            Least Squares   F-statistic:                     3297.
Date:                           Wed, 17 Jun 2026   Prob (F-statistic):          9.92e-220
Time:                                   14:43:27   Log-Likelihood:                -479.38
No. Observations:                            492   AIC:                             962.8
Df Residuals:                                490   BIC:                             971.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -3.3977      0.155    -21.877      0.000      -3.703      -3.093
SPY_Rolling_Future_Return_5y     8.9388      0.156     57.421      0.000       8.633       9.245
==============================================================================
Omnibus:                      107.575   Durbin-Watson:                   0.059
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              211.213
Skew:                          -1.205   Prob(JB):                     1.37e-46
Kurtosis:                       5.120   Cond. No.                         10.6
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Rolling Returns Following Drawdowns Deviation (SPY & UPRO) #

rolling_returns_positive_future_returns = pd.DataFrame(index=rolling_windows.keys(), data=rolling_windows.values())
rolling_returns_positive_future_returns.reset_index(inplace=True)
rolling_returns_positive_future_returns.rename(columns={"index":"Period", 0:"Days"}, inplace=True)

for drawdown in drawdown_levels:
    temp = rolling_returns_drawdown_stats.loc[rolling_returns_drawdown_stats["Drawdown"] == drawdown]
    temp = temp[["Period", "Positive_Future_Percentage"]]
    temp.rename(columns={"Positive_Future_Percentage" : f"Positive_Future_Percentage_Post_{drawdown}_Drawdown"}, inplace=True)
    rolling_returns_positive_future_returns = pd.merge(rolling_returns_positive_future_returns, temp, left_on="Period", right_on="Period", how="outer")
    rolling_returns_positive_future_returns.sort_values(by="Days", ascending=True, inplace=True)

rolling_returns_positive_future_returns.drop(columns={"Days"}, inplace=True)
rolling_returns_positive_future_returns.reset_index(drop=True, inplace=True)
pandas_set_decimal_places(2)
display(rolling_returns_positive_future_returns.set_index("Period"))

	Positive_Future_Percentage_Post_-0.1_Drawdown	Positive_Future_Percentage_Post_-0.2_Drawdown	Positive_Future_Percentage_Post_-0.3_Drawdown	Positive_Future_Percentage_Post_-0.4_Drawdown	Positive_Future_Percentage_Post_-0.5_Drawdown	Positive_Future_Percentage_Post_-0.6_Drawdown	Positive_Future_Percentage_Post_-0.7_Drawdown	Positive_Future_Percentage_Post_-0.8_Drawdown	Positive_Future_Percentage_Post_-0.9_Drawdown
Period
1d	0.54	0.54	0.54	0.54	0.54	0.55	0.55	0.55	0.55
1w	0.57	0.57	0.56	0.56	0.56	0.56	0.57	0.57	0.59
1m	0.63	0.62	0.61	0.61	0.62	0.62	0.62	0.65	0.67
3m	0.67	0.65	0.63	0.63	0.65	0.65	0.66	0.68	0.77
6m	0.70	0.69	0.68	0.67	0.69	0.70	0.73	0.75	0.79
1y	0.74	0.73	0.73	0.73	0.74	0.79	0.84	0.87	0.96
2y	0.77	0.78	0.78	0.78	0.77	0.81	0.92	0.99	1.00
3y	0.72	0.72	0.73	0.73	0.72	0.74	0.87	0.99	1.00
4y	0.69	0.69	0.68	0.69	0.68	0.71	0.81	1.00	1.00
5y	0.66	0.66	0.66	0.66	0.65	0.67	0.74	0.97	1.00

plot_scatter(
    df=rolling_returns_positive_future_returns,
    x_plot_column="Period",
    y_plot_columns=[col for col in rolling_returns_positive_future_returns.columns if col != "Period"],
    title="UPRO Future Return by Time Period Post Drawdown",
    x_label="Rolling Return Time Period",
    x_format="String",
    x_format_decimal_places=0,
    x_tick_spacing=1,
    x_tick_start=None,
    x_tick_rotation=0,
    y_label="Positive Future Return Percentage",
    y_format="Decimal",
    y_format_decimal_places=2,
    y_tick_spacing="Auto",
    y_tick_rotation=0,
    plot_OLS_regression_line=False,
    OLS_column=None,
    plot_Ridge_regression_line=False,
    Ridge_column=None,
    plot_RidgeCV_regression_line=False,
    RidgeCV_column=None,
    regression_constant=False,
    grid=True,
    legend=True,
    export_plot=False,
    plot_file_name=None,
)

png

This plot summarizes the future rolling returns well. Similar as to QQQ/TQQQ, for rolling returns up to ~3 months following all drawdown levels, we see the rolling returns of UPRO are positive ~65% of the time.

As we extend the time horizon, out to the 2y, 3y, 4y, and 5y mark, the percentage of positive rolling returns following an 80% drawdown increases significantly, and is greater than 95%. This suggests that while the volatility decay effect is present for UPRO, it may not be as severe as that of TQQQ, which could be due to the less extreme return profile of SPY compared to QQQ.

As an investor, this suggests that the optimal time to buy UPRO would be following a drawdown of 50% or more, and holding for at least 2 years. One could dollar cost average into UPRO following a drawdown of 50% or more, and continue to add to the position with a consistent contribution schedule until all capital has been allocated.

Future Investigation #

There are a couple of ideas for future investigation that would be interesting to explore:

Expand the analysis of SPY/UPRO to SPX/UPRO (using Bloomberg data for SPX), and extrapolate UPRO return data back to January of 1975.
Implement and backtest a strategy that DCA’s into UPRO on a consistent schdule (monthly, quarterly, etc.)

Code #

The Jupyter notebook with the functions and all other code is available here.
The HTML export of the jupyter notebook is available here.
The PDF export of the jupyter notebook is available here.