Leveraged ETF Return Dispersion

Introduction #

Over the past 15 years (or so), leveraged ETFs have become a frequently used vehicle for trading equity indices, sectors, and other asset classes for the investor that is seeking to use leverage to amply their returns. The question remains, however, what happens to the returns of leveraged ETFs over 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 amply the upside, 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 Directories To Path #

# 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_timeseries: Plot the timeseries data from a DataFrame for a specified date range and columns.
• run_linear_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_timeseries import plot_timeseries
from run_linear_regression import run_linear_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 file. Otherwise, we can download it from Yahoo Finance using the yf_pull_data function.

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"
})

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This gives us:

display(qqq)

And the plot of the timseries of ajdusted close prices:

plot_timeseries(
    df=qqq,
    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_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,
)

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"
})

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This gives us:

display(tqqq)

And the plot of the timseries of ajdusted close prices:

plot_timeseries(
    df=tqqq,
    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=1,
    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,
)

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)

And now the plot for the cumulative returns:

plot_timeseries(
    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_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,
)

And the drawdown plot:

plot_timeseries(
    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_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,
)

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 ~10% drawdown), 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 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)

Note that these statistics are being run on the partially-adjusted close prices, which are not the true returns, but they do give us a good 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 standard deviation is also much higher, which is consistent with the idea of leverage amplifying both the upside and the downside. 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 COVID.

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.

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_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,
)

model = run_linear_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:                 1.492e+06
Date:                Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                        14:26:00   Log-Likelihood:                 19405.
No. Observations:                4045   AIC:                        -3.881e+04
Df Residuals:                    4043   BIC:                        -3.879e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const      -8.554e-05   3.15e-05     -2.720      0.007      -0.000   -2.39e-05
QQQ_Return     2.9552      0.002   1221.329      0.000       2.950       2.960
==============================================================================
Omnibus:                     5272.508   Durbin-Watson:                   2.566
Prob(Omnibus):                  0.000   Jarque-Bera (JB):          9175082.049
Skew:                          -6.344   Prob(JB):                         0.00
Kurtosis:                     235.974   Cond. No.                         77.1
==============================================================================

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, we can say that TQQQ does in fact return ~3x QQQ. We would also intuitively expect the coefficient to be 0, which it is nearly.

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

Extrapolate Data (QQQ & TQQQ) #

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. 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[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
print(qqq_tqqq_extrap.loc["2010-02-08":"2010-02-13"])

            QQQ_Close  TQQQ_Close  QQQ_Return  TQQQ_Return
Date                                                      
2010-02-08  42.669998         NaN   -0.007213    -0.021315
2010-02-09  43.110001         NaN    0.010312     0.030473
2010-02-10  43.020000         NaN   -0.002088    -0.006169
2010-02-11  43.669998    0.216276    0.015109     0.044650
2010-02-12  43.759998    0.217161    0.002061     0.004092

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
print(qqq_tqqq_extrap.loc["2010-02-08":"2010-02-13"])

            QQQ_Close  TQQQ_Close  QQQ_Return  TQQQ_Return
Date                                                      
2010-02-08  42.669998    0.202157   -0.007213    -0.021315
2010-02-09  43.110001    0.208317    0.010312     0.030473
2010-02-10  43.020000    0.207032   -0.002088    -0.006169
2010-02-11  43.669998    0.216276    0.015109     0.044650
2010-02-12  43.759998    0.217161    0.002061     0.004092

And the complete DataFrame with the extrapolated values:

display(qqq_tqqq_extrap)

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_timeseries(
    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_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,
)

plot_timeseries(
    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=1,
    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,
)

plot_timeseries(
    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=1,
    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,
)

plot_timeseries(
    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=1,
    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,
)

Some quick comments before we look at rolling returns. The drawdown is nearly 100%… which 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 is so severe that it would be very difficult for any investor to hold through it.

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_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_linear_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[0]
    intercept_pvalue = model.pvalues[0]   # p-value for Intercept
    slope = model.params[1]
    slope_pvalue = model.pvalues[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],
        "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])

                              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:                 7.215e+06
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:03   Log-Likelihood:                 34352.
No. Observations:                    6794   AIC:                        -6.870e+04
Df Residuals:                        6792   BIC:                        -6.869e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                 -5.091e-05   1.87e-05     -2.721      0.007   -8.76e-05   -1.42e-05
QQQ_Rolling_Return_1d     2.9551      0.001   2686.049      0.000       2.953       2.957
==============================================================================
Omnibus:                    10188.882   Durbin-Watson:                   2.565
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         43894224.936
Skew:                          -8.279   Prob(JB):                         0.00
Kurtosis:                     396.425   Cond. No.                         58.8
==============================================================================

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

                              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:                 1.116e+06
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:04   Log-Likelihood:                 23152.
No. Observations:                    6790   AIC:                        -4.630e+04
Df Residuals:                        6788   BIC:                        -4.629e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0008   9.73e-05     -8.330      0.000      -0.001      -0.001
QQQ_Rolling_Return_1w     2.9524      0.003   1056.353      0.000       2.947       2.958
==============================================================================
Omnibus:                     2839.003   Durbin-Watson:                   0.932
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           563907.961
Skew:                          -0.863   Prob(JB):                         0.00
Kurtosis:                      47.612   Cond. No.                         28.8
==============================================================================

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

                              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:                 3.695e+05
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:05   Log-Likelihood:                 14852.
No. Observations:                    6774   AIC:                        -2.970e+04
Df Residuals:                        6772   BIC:                        -2.969e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0037      0.000    -11.064      0.000      -0.004      -0.003
QQQ_Rolling_Return_1m     2.9305      0.005    607.853      0.000       2.921       2.940
==============================================================================
Omnibus:                     1627.646   Durbin-Watson:                   0.296
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            69016.055
Skew:                           0.357   Prob(JB):                         0.00
Kurtosis:                      18.621   Cond. No.                         14.7
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_3m   R-squared:                       0.958
Model:                                OLS   Adj. R-squared:                  0.958
Method:                     Least Squares   F-statistic:                 1.549e+05
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:06   Log-Likelihood:                 7946.0
No. Observations:                    6732   AIC:                        -1.589e+04
Df Residuals:                        6730   BIC:                        -1.587e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0083      0.001     -8.841      0.000      -0.010      -0.006
QQQ_Rolling_Return_3m     2.9848      0.008    393.626      0.000       2.970       3.000
==============================================================================
Omnibus:                     3461.856   Durbin-Watson:                   0.105
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            79524.170
Skew:                           1.962   Prob(JB):                         0.00
Kurtosis:                      19.374   Cond. No.                         8.38
==============================================================================

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

                              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:                 7.247e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:08   Log-Likelihood:                 2610.3
No. Observations:                    6669   AIC:                            -5217.
Df Residuals:                        6667   BIC:                            -5203.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0096      0.002     -4.515      0.000      -0.014      -0.005
QQQ_Rolling_Return_6m     3.0396      0.011    269.206      0.000       3.017       3.062
==============================================================================
Omnibus:                     3654.453   Durbin-Watson:                   0.056
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            60093.604
Skew:                           2.262   Prob(JB):                         0.00
Kurtosis:                      16.993   Cond. No.                         5.66
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_1y   R-squared:                       0.880
Model:                                OLS   Adj. R-squared:                  0.880
Method:                     Least Squares   F-statistic:                 4.785e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:09   Log-Likelihood:                -892.54
No. Observations:                    6543   AIC:                             1789.
Df Residuals:                        6541   BIC:                             1803.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                     0.0191      0.004      5.035      0.000       0.012       0.026
QQQ_Rolling_Return_1y     2.8375      0.013    218.746      0.000       2.812       2.863
==============================================================================
Omnibus:                     3492.635   Durbin-Watson:                   0.037
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            68111.624
Skew:                           2.122   Prob(JB):                         0.00
Kurtosis:                      18.226   Cond. No.                         3.84
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_2y   R-squared:                       0.849
Model:                                OLS   Adj. R-squared:                  0.848
Method:                     Least Squares   F-statistic:                 3.522e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:10   Log-Likelihood:                -4421.2
No. Observations:                    6291   AIC:                             8846.
Df Residuals:                        6289   BIC:                             8860.
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                     0.0096      0.007      1.294      0.196      -0.005       0.024
QQQ_Rolling_Return_2y     3.1314      0.017    187.680      0.000       3.099       3.164
==============================================================================
Omnibus:                     1595.143   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4144.668
Skew:                           1.367   Prob(JB):                         0.00
Kurtosis:                       5.887   Cond. No.                         2.89
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_3y   R-squared:                       0.804
Model:                                OLS   Adj. R-squared:                  0.804
Method:                     Least Squares   F-statistic:                 2.481e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:11   Log-Likelihood:                -6691.5
No. Observations:                    6039   AIC:                         1.339e+04
Df Residuals:                        6037   BIC:                         1.340e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.0599      0.013     -4.763      0.000      -0.085      -0.035
QQQ_Rolling_Return_3y     3.3321      0.021    157.522      0.000       3.291       3.374
==============================================================================
Omnibus:                      855.515   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1466.078
Skew:                           0.939   Prob(JB):                         0.00
Kurtosis:                       4.516   Cond. No.                         2.66
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_4y   R-squared:                       0.781
Model:                                OLS   Adj. R-squared:                  0.781
Method:                     Least Squares   F-statistic:                 2.068e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:12   Log-Likelihood:                -8782.4
No. Observations:                    5787   AIC:                         1.757e+04
Df Residuals:                        5785   BIC:                         1.758e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.2922      0.021    -13.668      0.000      -0.334      -0.250
QQQ_Rolling_Return_4y     3.9343      0.027    143.797      0.000       3.881       3.988
==============================================================================
Omnibus:                      198.061   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              103.519
Skew:                           0.140   Prob(JB):                     3.32e-23
Kurtosis:                       2.407   Cond. No.                         2.66
==============================================================================

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

                              OLS Regression Results                              
==================================================================================
Dep. Variable:     TQQQ_Rolling_Return_5y   R-squared:                       0.743
Model:                                OLS   Adj. R-squared:                  0.743
Method:                     Least Squares   F-statistic:                 1.599e+04
Date:                    Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                            14:26:13   Log-Likelihood:                -12071.
No. Observations:                    5535   AIC:                         2.415e+04
Df Residuals:                        5533   BIC:                         2.416e+04
Df Model:                               1                                         
Covariance Type:                nonrobust                                         
=========================================================================================
                            coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
const                    -0.8853      0.044    -19.926      0.000      -0.972      -0.798
QQQ_Rolling_Return_5y     5.2054      0.041    126.462      0.000       5.125       5.286
==============================================================================
Omnibus:                      314.365   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              460.149
Skew:                           0.498   Prob(JB):                    1.20e-100
Kurtosis:                       4.002   Cond. No.                         2.73
==============================================================================

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 -0.30x 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 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

display(rolling_returns_stats)

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_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,
)

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_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,
)

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_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,
)

pandas_set_decimal_places(3)
display(rolling_returns_stats.set_index("Period"))

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 over the past 15 years - 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]

# 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_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_linear_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[0]
            # intercept_pvalue = model.pvalues[0]   # p-value for Intercept
            slope = model.params[1]
            # slope_pvalue = model.pvalues[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.")

                                  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:                 6.825e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:15   Log-Likelihood:                 33542.
No. Observations:                           6648   AIC:                        -6.708e+04
Df Residuals:                               6646   BIC:                        -6.707e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.203e-05   1.91e-05     -2.721      0.007   -8.95e-05   -1.45e-05
QQQ_Rolling_Future_Return_1d     2.9551      0.001   2612.377      0.000       2.953       2.957
==============================================================================
Omnibus:                     9913.687   Durbin-Watson:                   2.565
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         41109927.245
Skew:                          -8.187   Prob(JB):                         0.00
Kurtosis:                     387.894   Cond. No.                         59.2
==============================================================================

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

                                  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:                 1.088e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:16   Log-Likelihood:                 22709.
No. Observations:                           6644   AIC:                        -4.541e+04
Df Residuals:                               6642   BIC:                        -4.540e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.76e-05     -8.221      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9526      0.003   1043.242      0.000       2.947       2.958
==============================================================================
Omnibus:                     2701.400   Durbin-Watson:                   0.939
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           586368.143
Skew:                          -0.779   Prob(JB):                         0.00
Kurtosis:                      48.997   Cond. No.                         29.1
==============================================================================

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

                                  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:                 3.703e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:17   Log-Likelihood:                 14696.
No. Observations:                           6628   AIC:                        -2.939e+04
Df Residuals:                               6626   BIC:                        -2.937e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.530      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9304      0.005    608.513      0.000       2.921       2.940
==============================================================================
Omnibus:                     1679.490   Durbin-Watson:                   0.312
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            80862.122
Skew:                           0.393   Prob(JB):                         0.00
Kurtosis:                      20.093   Cond. No.                         14.9
==============================================================================

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

                                  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:                 1.458e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:18   Log-Likelihood:                 7947.5
No. Observations:                           6586   AIC:                        -1.589e+04
Df Residuals:                               6584   BIC:                        -1.588e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0076      0.001     -8.260      0.000      -0.009      -0.006
QQQ_Rolling_Future_Return_3m     2.9582      0.008    381.884      0.000       2.943       2.973
==============================================================================
Omnibus:                     3401.457   Durbin-Watson:                   0.113
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            86281.946
Skew:                           1.942   Prob(JB):                         0.00
Kurtosis:                      20.301   Cond. No.                         8.69
==============================================================================

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

                                  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:                 7.566e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:20   Log-Likelihood:                 3154.5
No. Observations:                           6523   AIC:                            -6305.
Df Residuals:                               6521   BIC:                            -6291.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0042      0.002     -2.132      0.033      -0.008      -0.000
QQQ_Rolling_Future_Return_6m     2.9626      0.011    275.060      0.000       2.941       2.984
==============================================================================
Omnibus:                     4154.198   Durbin-Watson:                   0.065
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           100612.082
Skew:                           2.647   Prob(JB):                         0.00
Kurtosis:                      21.498   Cond. No.                         5.85
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.893
Model:                                       OLS   Adj. R-squared:                  0.893
Method:                            Least Squares   F-statistic:                 5.326e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:21   Log-Likelihood:                -135.51
No. Observations:                           6397   AIC:                             275.0
Df Residuals:                               6395   BIC:                             288.6
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0228      0.003      6.654      0.000       0.016       0.030
QQQ_Rolling_Future_Return_1y     2.8089      0.012    230.784      0.000       2.785       2.833
==============================================================================
Omnibus:                     2631.338   Durbin-Watson:                   0.052
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            29220.801
Skew:                           1.657   Prob(JB):                         0.00
Kurtosis:                      12.932   Cond. No.                         4.00
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.848
Model:                                       OLS   Adj. R-squared:                  0.848
Method:                            Least Squares   F-statistic:                 3.426e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:22   Log-Likelihood:                -4260.2
No. Observations:                           6145   AIC:                             8524.
Df Residuals:                               6143   BIC:                             8538.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0188      0.008     -2.463      0.014      -0.034      -0.004
QQQ_Rolling_Future_Return_2y     3.2014      0.017    185.083      0.000       3.167       3.235
==============================================================================
Omnibus:                     1670.286   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4641.855
Skew:                           1.436   Prob(JB):                         0.00
Kurtosis:                       6.143   Cond. No.                         3.02
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.811
Model:                                       OLS   Adj. R-squared:                  0.811
Method:                            Least Squares   F-statistic:                 2.522e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:23   Log-Likelihood:                -6367.9
No. Observations:                           5893   AIC:                         1.274e+04
Df Residuals:                               5891   BIC:                         1.275e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1590      0.013    -12.136      0.000      -0.185      -0.133
QQQ_Rolling_Future_Return_3y     3.5025      0.022    158.807      0.000       3.459       3.546
==============================================================================
Omnibus:                      868.866   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1529.067
Skew:                           0.959   Prob(JB):                         0.00
Kurtosis:                       4.595   Cond. No.                         2.86
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.783
Model:                                       OLS   Adj. R-squared:                  0.783
Method:                            Least Squares   F-statistic:                 2.039e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:24   Log-Likelihood:                -8491.0
No. Observations:                           5641   AIC:                         1.699e+04
Df Residuals:                               5639   BIC:                         1.700e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.4268      0.023    -18.906      0.000      -0.471      -0.383
QQQ_Rolling_Future_Return_4y     4.0976      0.029    142.777      0.000       4.041       4.154
==============================================================================
Omnibus:                      125.491   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               78.475
Skew:                           0.148   Prob(JB):                     9.11e-18
Kurtosis:                       2.504   Cond. No.                         2.85
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.745
Model:                                       OLS   Adj. R-squared:                  0.745
Method:                            Least Squares   F-statistic:                 1.578e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:25   Log-Likelihood:                -11719.
No. Observations:                           5389   AIC:                         2.344e+04
Df Residuals:                               5387   BIC:                         2.345e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.1137      0.047    -23.759      0.000      -1.206      -1.022
QQQ_Rolling_Future_Return_5y     5.3970      0.043    125.610      0.000       5.313       5.481
==============================================================================
Omnibus:                      275.950   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              406.209
Skew:                           0.459   Prob(JB):                     6.21e-89
Kurtosis:                       3.983   Cond. No.                         2.90
==============================================================================

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

                                  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:                 6.602e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:27   Log-Likelihood:                 33115.
No. Observations:                           6571   AIC:                        -6.623e+04
Df Residuals:                               6569   BIC:                        -6.621e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.264e-05   1.93e-05     -2.721      0.007   -9.06e-05   -1.47e-05
QQQ_Rolling_Future_Return_1d     2.9551      0.001   2569.475      0.000       2.953       2.957
==============================================================================
Omnibus:                     9769.052   Durbin-Watson:                   2.565
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         39689255.457
Skew:                          -8.139   Prob(JB):                         0.00
Kurtosis:                     383.390   Cond. No.                         59.5
==============================================================================

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

                                  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:                 1.069e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:28   Log-Likelihood:                 22453.
No. Observations:                           6567   AIC:                        -4.490e+04
Df Residuals:                               6565   BIC:                        -4.489e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008    9.8e-05     -8.016      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9517      0.003   1033.957      0.000       2.946       2.957
==============================================================================
Omnibus:                     2705.152   Durbin-Watson:                   0.933
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           594917.588
Skew:                          -0.803   Prob(JB):                         0.00
Kurtosis:                      49.601   Cond. No.                         29.2
==============================================================================

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

                                  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:                 3.703e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:29   Log-Likelihood:                 14653.
No. Observations:                           6551   AIC:                        -2.930e+04
Df Residuals:                               6549   BIC:                        -2.929e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0035      0.000    -10.801      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9224      0.005    608.511      0.000       2.913       2.932
==============================================================================
Omnibus:                     1535.850   Durbin-Watson:                   0.317
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            81019.960
Skew:                           0.160   Prob(JB):                         0.00
Kurtosis:                      20.226   Cond. No.                         15.0
==============================================================================

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

                                  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:                 1.544e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:30   Log-Likelihood:                 8321.1
No. Observations:                           6509   AIC:                        -1.664e+04
Df Residuals:                               6507   BIC:                        -1.662e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0070      0.001     -8.175      0.000      -0.009      -0.005
QQQ_Rolling_Future_Return_3m     2.9103      0.007    392.952      0.000       2.896       2.925
==============================================================================
Omnibus:                     2145.969   Durbin-Watson:                   0.136
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            38765.286
Skew:                           1.109   Prob(JB):                         0.00
Kurtosis:                      14.748   Cond. No.                         8.87
==============================================================================

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

                                  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:                 8.102e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:32   Log-Likelihood:                 3779.1
No. Observations:                           6446   AIC:                            -7554.
Df Residuals:                               6444   BIC:                            -7541.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0015      0.002     -0.860      0.390      -0.005       0.002
QQQ_Rolling_Future_Return_6m     2.8743      0.010    284.646      0.000       2.855       2.894
==============================================================================
Omnibus:                     3183.205   Durbin-Watson:                   0.077
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            49759.451
Skew:                           1.977   Prob(JB):                         0.00
Kurtosis:                      16.024   Cond. No.                         6.04
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.898
Model:                                       OLS   Adj. R-squared:                  0.898
Method:                            Least Squares   F-statistic:                 5.551e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:33   Log-Likelihood:                 115.86
No. Observations:                           6320   AIC:                            -227.7
Df Residuals:                               6318   BIC:                            -214.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0227      0.003      6.836      0.000       0.016       0.029
QQQ_Rolling_Future_Return_1y     2.8344      0.012    235.614      0.000       2.811       2.858
==============================================================================
Omnibus:                     2422.820   Durbin-Watson:                   0.068
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            19442.842
Skew:                           1.621   Prob(JB):                         0.00
Kurtosis:                      10.958   Cond. No.                         4.09
==============================================================================

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

                                  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:                 3.365e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:34   Log-Likelihood:                -4181.2
No. Observations:                           6068   AIC:                             8366.
Df Residuals:                               6066   BIC:                             8380.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0288      0.008     -3.698      0.000      -0.044      -0.014
QQQ_Rolling_Future_Return_2y     3.2276      0.018    183.436      0.000       3.193       3.262
==============================================================================
Omnibus:                     1695.729   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4868.218
Skew:                           1.463   Prob(JB):                         0.00
Kurtosis:                       6.270   Cond. No.                         3.07
==============================================================================

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

                                  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:                 2.547e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:35   Log-Likelihood:                -6191.6
No. Observations:                           5816   AIC:                         1.239e+04
Df Residuals:                               5814   BIC:                         1.240e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2170      0.013    -16.189      0.000      -0.243      -0.191
QQQ_Rolling_Future_Return_3y     3.6008      0.023    159.606      0.000       3.557       3.645
==============================================================================
Omnibus:                      874.268   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1570.330
Skew:                           0.966   Prob(JB):                         0.00
Kurtosis:                       4.657   Cond. No.                         2.98
==============================================================================

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

                                  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:                 2.024e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:36   Log-Likelihood:                -8333.7
No. Observations:                           5564   AIC:                         1.667e+04
Df Residuals:                               5562   BIC:                         1.668e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5059      0.023    -21.736      0.000      -0.552      -0.460
QQQ_Rolling_Future_Return_4y     4.1927      0.029    142.267      0.000       4.135       4.250
==============================================================================
Omnibus:                       89.406   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               63.314
Skew:                           0.153   Prob(JB):                     1.78e-14
Kurtosis:                       2.576   Cond. No.                         2.96
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.747
Model:                                       OLS   Adj. R-squared:                  0.747
Method:                            Least Squares   F-statistic:                 1.566e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:37   Log-Likelihood:                -11532.
No. Observations:                           5312   AIC:                         2.307e+04
Df Residuals:                               5310   BIC:                         2.308e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.2438      0.048    -25.759      0.000      -1.338      -1.149
QQQ_Rolling_Future_Return_5y     5.5050      0.044    125.134      0.000       5.419       5.591
==============================================================================
Omnibus:                      256.613   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              381.078
Skew:                           0.437   Prob(JB):                     1.78e-83
Kurtosis:                       3.979   Cond. No.                         3.00
==============================================================================

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

                                  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:                 6.433e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:39   Log-Likelihood:                 32894.
No. Observations:                           6531   AIC:                        -6.578e+04
Df Residuals:                               6529   BIC:                        -6.577e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.296e-05   1.95e-05     -2.721      0.007   -9.11e-05   -1.48e-05
QQQ_Rolling_Future_Return_1d     2.9551      0.001   2536.404      0.000       2.953       2.957
==============================================================================
Omnibus:                     9694.083   Durbin-Watson:                   2.565
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         38964571.195
Skew:                          -8.113   Prob(JB):                         0.00
Kurtosis:                     381.052   Cond. No.                         59.9
==============================================================================

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

                                  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:                 1.119e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:40   Log-Likelihood:                 22520.
No. Observations:                           6527   AIC:                        -4.504e+04
Df Residuals:                               6525   BIC:                        -4.502e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.53e-05     -8.333      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9552      0.003   1057.792      0.000       2.950       2.961
==============================================================================
Omnibus:                     3463.566   Durbin-Watson:                   0.902
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           388402.497
Skew:                          -1.580   Prob(JB):                         0.00
Kurtosis:                      40.659   Cond. No.                         29.4
==============================================================================

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

                                  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:                 3.675e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:41   Log-Likelihood:                 14623.
No. Observations:                           6511   AIC:                        -2.924e+04
Df Residuals:                               6509   BIC:                        -2.923e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0035      0.000    -11.033      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9176      0.005    606.191      0.000       2.908       2.927
==============================================================================
Omnibus:                     1511.925   Durbin-Watson:                   0.308
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            81855.191
Skew:                           0.083   Prob(JB):                         0.00
Kurtosis:                      20.369   Cond. No.                         15.2
==============================================================================

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

                                  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:                 1.585e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:42   Log-Likelihood:                 8457.5
No. Observations:                           6469   AIC:                        -1.691e+04
Df Residuals:                               6467   BIC:                        -1.690e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0067      0.001     -8.040      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8943      0.007    398.066      0.000       2.880       2.909
==============================================================================
Omnibus:                     1346.125   Durbin-Watson:                   0.103
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15267.066
Skew:                           0.666   Prob(JB):                         0.00
Kurtosis:                      10.407   Cond. No.                         8.94
==============================================================================

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

                                  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:                 8.378e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:43   Log-Likelihood:                 4128.3
No. Observations:                           6406   AIC:                            -8253.
Df Residuals:                               6404   BIC:                            -8239.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.002     -0.337      0.736      -0.004       0.003
QQQ_Rolling_Future_Return_6m     2.8222      0.010    289.447      0.000       2.803       2.841
==============================================================================
Omnibus:                     2682.468   Durbin-Watson:                   0.109
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            36485.982
Skew:                           1.629   Prob(JB):                         0.00
Kurtosis:                      14.228   Cond. No.                         6.16
==============================================================================

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

                                  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:                 5.754e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:45   Log-Likelihood:                 291.84
No. Observations:                           6280   AIC:                            -579.7
Df Residuals:                               6278   BIC:                            -566.2
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0224      0.003      6.919      0.000       0.016       0.029
QQQ_Rolling_Future_Return_1y     2.8516      0.012    239.879      0.000       2.828       2.875
==============================================================================
Omnibus:                     1986.661   Durbin-Watson:                   0.043
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8583.652
Skew:                           1.493   Prob(JB):                         0.00
Kurtosis:                       7.887   Cond. No.                         4.14
==============================================================================

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

                                  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:                 3.331e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:46   Log-Likelihood:                -4141.4
No. Observations:                           6028   AIC:                             8287.
Df Residuals:                               6026   BIC:                             8300.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0317      0.008     -4.039      0.000      -0.047      -0.016
QQQ_Rolling_Future_Return_2y     3.2366      0.018    182.524      0.000       3.202       3.271
==============================================================================
Omnibus:                     1703.262   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4959.569
Skew:                           1.473   Prob(JB):                         0.00
Kurtosis:                       6.326   Cond. No.                         3.10
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.816
Model:                                       OLS   Adj. R-squared:                  0.816
Method:                            Least Squares   F-statistic:                 2.566e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:47   Log-Likelihood:                -6094.3
No. Observations:                           5776   AIC:                         1.219e+04
Df Residuals:                               5774   BIC:                         1.221e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2502      0.014    -18.446      0.000      -0.277      -0.224
QQQ_Rolling_Future_Return_3y     3.6568      0.023    160.193      0.000       3.612       3.702
==============================================================================
Omnibus:                      876.215   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1596.509
Skew:                           0.967   Prob(JB):                         0.00
Kurtosis:                       4.700   Cond. No.                         3.05
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.785
Model:                                       OLS   Adj. R-squared:                  0.785
Method:                            Least Squares   F-statistic:                 2.017e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:48   Log-Likelihood:                -8250.6
No. Observations:                           5524   AIC:                         1.651e+04
Df Residuals:                               5522   BIC:                         1.652e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5497      0.024    -23.230      0.000      -0.596      -0.503
QQQ_Rolling_Future_Return_4y     4.2450      0.030    142.023      0.000       4.186       4.304
==============================================================================
Omnibus:                       72.624   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               55.217
Skew:                           0.155   Prob(JB):                     1.02e-12
Kurtosis:                       2.620   Cond. No.                         3.02
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.748
Model:                                       OLS   Adj. R-squared:                  0.747
Method:                            Least Squares   F-statistic:                 1.560e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:49   Log-Likelihood:                -11434.
No. Observations:                           5272   AIC:                         2.287e+04
Df Residuals:                               5270   BIC:                         2.288e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3156      0.049    -26.822      0.000      -1.412      -1.219
QQQ_Rolling_Future_Return_5y     5.5645      0.045    124.913      0.000       5.477       5.652
==============================================================================
Omnibus:                      246.317   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              368.513
Skew:                           0.424   Prob(JB):                     9.51e-81
Kurtosis:                       3.980   Cond. No.                         3.05
==============================================================================

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

                                  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:                 6.385e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:50   Log-Likelihood:                 32833.
No. Observations:                           6520   AIC:                        -6.566e+04
Df Residuals:                               6518   BIC:                        -6.565e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -5.305e-05   1.95e-05     -2.721      0.007   -9.13e-05   -1.48e-05
QQQ_Rolling_Future_Return_1d     2.9551      0.001   2526.915      0.000       2.953       2.957
==============================================================================
Omnibus:                     9673.351   Durbin-Watson:                   2.565
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         38764591.768
Skew:                          -8.106   Prob(JB):                         0.00
Kurtosis:                     380.398   Cond. No.                         60.0
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:52   Log-Likelihood:                 22527.
No. Observations:                           6516   AIC:                        -4.505e+04
Df Residuals:                               6514   BIC:                        -4.504e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.47e-05     -8.566      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9535      0.003   1057.421      0.000       2.948       2.959
==============================================================================
Omnibus:                     3593.998   Durbin-Watson:                   0.881
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           403875.873
Skew:                          -1.687   Prob(JB):                         0.00
Kurtosis:                      41.421   Cond. No.                         29.6
==============================================================================

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

                                  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:                 3.661e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:53   Log-Likelihood:                 14619.
No. Observations:                           6500   AIC:                        -2.923e+04
Df Residuals:                               6498   BIC:                        -2.922e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0035      0.000    -11.021      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9150      0.005    605.092      0.000       2.906       2.924
==============================================================================
Omnibus:                     1510.573   Durbin-Watson:                   0.297
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            82826.988
Skew:                           0.062   Prob(JB):                         0.00
Kurtosis:                      20.487   Cond. No.                         15.2
==============================================================================

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

                                  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:                 1.591e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:54   Log-Likelihood:                 8471.5
No. Observations:                           6458   AIC:                        -1.694e+04
Df Residuals:                               6456   BIC:                        -1.693e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0065      0.001     -7.870      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8924      0.007    398.870      0.000       2.878       2.907
==============================================================================
Omnibus:                     1376.418   Durbin-Watson:                   0.101
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15591.855
Skew:                           0.692   Prob(JB):                         0.00
Kurtosis:                      10.485   Cond. No.                         8.95
==============================================================================

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

                                  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:                 8.656e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:55   Log-Likelihood:                 4305.0
No. Observations:                           6395   AIC:                            -8606.
Df Residuals:                               6393   BIC:                            -8593.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -4.855e-06      0.002     -0.003      0.998      -0.003       0.003
QQQ_Rolling_Future_Return_6m     2.8036      0.010    294.207      0.000       2.785       2.822
==============================================================================
Omnibus:                     1637.994   Durbin-Watson:                   0.057
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8138.991
Skew:                           1.146   Prob(JB):                         0.00
Kurtosis:                       8.029   Cond. No.                         6.19
==============================================================================

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

                                  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:                 5.795e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:57   Log-Likelihood:                 326.09
No. Observations:                           6269   AIC:                            -648.2
Df Residuals:                               6267   BIC:                            -634.7
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0211      0.003      6.535      0.000       0.015       0.027
QQQ_Rolling_Future_Return_1y     2.8618      0.012    240.727      0.000       2.838       2.885
==============================================================================
Omnibus:                     1984.175   Durbin-Watson:                   0.039
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8573.296
Skew:                           1.494   Prob(JB):                         0.00
Kurtosis:                       7.888   Cond. No.                         4.16
==============================================================================

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

                                  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:                 3.333e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:58   Log-Likelihood:                -4121.6
No. Observations:                           6017   AIC:                             8247.
Df Residuals:                               6015   BIC:                             8261.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0345      0.008     -4.386      0.000      -0.050      -0.019
QQQ_Rolling_Future_Return_2y     3.2439      0.018    182.575      0.000       3.209       3.279
==============================================================================
Omnibus:                     1712.301   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5037.509
Skew:                           1.480   Prob(JB):                         0.00
Kurtosis:                       6.367   Cond. No.                         3.11
==============================================================================

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

                                  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:                 2.581e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:26:59   Log-Likelihood:                -6058.9
No. Observations:                           5765   AIC:                         1.212e+04
Df Residuals:                               5763   BIC:                         1.214e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2618      0.014    -19.253      0.000      -0.288      -0.235
QQQ_Rolling_Future_Return_3y     3.6767      0.023    160.644      0.000       3.632       3.722
==============================================================================
Omnibus:                      869.452   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1585.916
Skew:                           0.962   Prob(JB):                         0.00
Kurtosis:                       4.703   Cond. No.                         3.07
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.786
Model:                                       OLS   Adj. R-squared:                  0.786
Method:                            Least Squares   F-statistic:                 2.019e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:00   Log-Likelihood:                -8223.4
No. Observations:                           5513   AIC:                         1.645e+04
Df Residuals:                               5511   BIC:                         1.646e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5646      0.024    -23.758      0.000      -0.611      -0.518
QQQ_Rolling_Future_Return_4y     4.2630      0.030    142.094      0.000       4.204       4.322
==============================================================================
Omnibus:                       67.431   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               52.172
Skew:                           0.153   Prob(JB):                     4.69e-12
Kurtosis:                       2.634   Cond. No.                         3.04
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.748
Model:                                       OLS   Adj. R-squared:                  0.748
Method:                            Least Squares   F-statistic:                 1.562e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:01   Log-Likelihood:                -11403.
No. Observations:                           5261   AIC:                         2.281e+04
Df Residuals:                               5259   BIC:                         2.282e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3405      0.049    -27.216      0.000      -1.437      -1.244
QQQ_Rolling_Future_Return_5y     5.5855      0.045    124.962      0.000       5.498       5.673
==============================================================================
Omnibus:                      241.818   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              363.534
Skew:                           0.417   Prob(JB):                     1.15e-79
Kurtosis:                       3.981   Cond. No.                         3.07
==============================================================================

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

                                  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:                 6.255e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:03   Log-Likelihood:                 32495.
No. Observations:                           6457   AIC:                        -6.499e+04
Df Residuals:                               6455   BIC:                        -6.497e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -4.913e-05   1.97e-05     -2.500      0.012   -8.77e-05   -1.06e-05
QQQ_Rolling_Future_Return_1d     2.9549      0.001   2501.003      0.000       2.953       2.957
==============================================================================
Omnibus:                     9580.111   Durbin-Watson:                   2.567
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         38114515.631
Skew:                          -8.108   Prob(JB):                         0.00
Kurtosis:                     379.038   Cond. No.                         60.1
==============================================================================

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

                                  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:                 1.109e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:04   Log-Likelihood:                 22311.
No. Observations:                           6453   AIC:                        -4.462e+04
Df Residuals:                               6451   BIC:                        -4.461e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0008   9.51e-05     -8.169      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9529      0.003   1053.321      0.000       2.947       2.958
==============================================================================
Omnibus:                     3540.518   Durbin-Watson:                   0.887
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           402393.065
Skew:                          -1.669   Prob(JB):                         0.00
Kurtosis:                      41.541   Cond. No.                         29.5
==============================================================================

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

                                  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:                 3.615e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:05   Log-Likelihood:                 14455.
No. Observations:                           6437   AIC:                        -2.891e+04
Df Residuals:                               6435   BIC:                        -2.889e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0034      0.000    -10.594      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9144      0.005    601.219      0.000       2.905       2.924
==============================================================================
Omnibus:                     1490.690   Durbin-Watson:                   0.295
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            81043.655
Skew:                           0.049   Prob(JB):                         0.00
Kurtosis:                      20.383   Cond. No.                         15.2
==============================================================================

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

                                  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:                 1.589e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:06   Log-Likelihood:                 8428.1
No. Observations:                           6424   AIC:                        -1.685e+04
Df Residuals:                               6422   BIC:                        -1.684e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0063      0.001     -7.529      0.000      -0.008      -0.005
QQQ_Rolling_Future_Return_3m     2.8919      0.007    398.560      0.000       2.878       2.906
==============================================================================
Omnibus:                     1375.718   Durbin-Watson:                   0.101
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15496.490
Skew:                           0.698   Prob(JB):                         0.00
Kurtosis:                      10.480   Cond. No.                         8.93
==============================================================================

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

                                  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:                 8.673e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:07   Log-Likelihood:                 4311.8
No. Observations:                           6392   AIC:                            -8620.
Df Residuals:                               6390   BIC:                            -8606.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0001      0.002      0.068      0.946      -0.003       0.003
QQQ_Rolling_Future_Return_6m     2.8037      0.010    294.503      0.000       2.785       2.822
==============================================================================
Omnibus:                     1653.772   Durbin-Watson:                   0.055
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8144.760
Skew:                           1.162   Prob(JB):                         0.00
Kurtosis:                       8.018   Cond. No.                         6.19
==============================================================================

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

                                  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:                 5.801e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:09   Log-Likelihood:                 333.68
No. Observations:                           6266   AIC:                            -663.4
Df Residuals:                               6264   BIC:                            -649.9
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     2.8644      0.012    240.863      0.000       2.841       2.888
==============================================================================
Omnibus:                     1982.919   Durbin-Watson:                   0.038
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8590.359
Skew:                           1.493   Prob(JB):                         0.00
Kurtosis:                       7.898   Cond. No.                         4.16
==============================================================================

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

                                  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:                 3.336e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:10   Log-Likelihood:                -4114.4
No. Observations:                           6014   AIC:                             8233.
Df Residuals:                               6012   BIC:                             8246.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0360      0.008     -4.571      0.000      -0.051      -0.021
QQQ_Rolling_Future_Return_2y     3.2475      0.018    182.648      0.000       3.213       3.282
==============================================================================
Omnibus:                     1717.522   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5077.382
Skew:                           1.483   Prob(JB):                         0.00
Kurtosis:                       6.386   Cond. No.                         3.12
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.818
Model:                                       OLS   Adj. R-squared:                  0.818
Method:                            Least Squares   F-statistic:                 2.588e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:11   Log-Likelihood:                -6046.1
No. Observations:                           5762   AIC:                         1.210e+04
Df Residuals:                               5760   BIC:                         1.211e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2659      0.014    -19.547      0.000      -0.293      -0.239
QQQ_Rolling_Future_Return_3y     3.6838      0.023    160.873      0.000       3.639       3.729
==============================================================================
Omnibus:                      866.163   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1580.248
Skew:                           0.959   Prob(JB):                         0.00
Kurtosis:                       4.704   Cond. No.                         3.08
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.786
Model:                                       OLS   Adj. R-squared:                  0.786
Method:                            Least Squares   F-statistic:                 2.021e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:12   Log-Likelihood:                -8214.5
No. Observations:                           5510   AIC:                         1.643e+04
Df Residuals:                               5508   BIC:                         1.645e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.5696      0.024    -23.943      0.000      -0.616      -0.523
QQQ_Rolling_Future_Return_4y     4.2692      0.030    142.161      0.000       4.210       4.328
==============================================================================
Omnibus:                       65.711   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               51.050
Skew:                           0.151   Prob(JB):                     8.22e-12
Kurtosis:                       2.638   Cond. No.                         3.05
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.748
Model:                                       OLS   Adj. R-squared:                  0.748
Method:                            Least Squares   F-statistic:                 1.563e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:13   Log-Likelihood:                -11393.
No. Observations:                           5258   AIC:                         2.279e+04
Df Residuals:                               5256   BIC:                         2.280e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.3493      0.049    -27.365      0.000      -1.446      -1.253
QQQ_Rolling_Future_Return_5y     5.5930      0.045    125.018      0.000       5.505       5.681
==============================================================================
Omnibus:                      239.977   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              361.637
Skew:                           0.414   Prob(JB):                     2.96e-79
Kurtosis:                       3.982   Cond. No.                         3.08
==============================================================================

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

                                  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:                 5.999e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:15   Log-Likelihood:                 31532.
No. Observations:                           6280   AIC:                        -6.306e+04
Df Residuals:                               6278   BIC:                        -6.305e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -4.232e-05   2.02e-05     -2.100      0.036   -8.18e-05   -2.81e-06
QQQ_Rolling_Future_Return_1d     2.9547      0.001   2449.263      0.000       2.952       2.957
==============================================================================
Omnibus:                     9286.532   Durbin-Watson:                   2.569
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         35696636.910
Skew:                          -8.058   Prob(JB):                         0.00
Kurtosis:                     371.999   Cond. No.                         59.9
==============================================================================

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

                                  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:                 1.076e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:16   Log-Likelihood:                 21675.
No. Observations:                           6280   AIC:                        -4.335e+04
Df Residuals:                               6278   BIC:                        -4.333e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0007    9.7e-05     -7.551      0.000      -0.001      -0.001
QQQ_Rolling_Future_Return_1w     2.9525      0.003   1037.392      0.000       2.947       2.958
==============================================================================
Omnibus:                     3408.626   Durbin-Watson:                   0.895
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           384667.664
Skew:                          -1.639   Prob(JB):                         0.00
Kurtosis:                      41.201   Cond. No.                         29.4
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:17   Log-Likelihood:                 14068.
No. Observations:                           6280   AIC:                        -2.813e+04
Df Residuals:                               6278   BIC:                        -2.812e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0032      0.000     -9.790      0.000      -0.004      -0.003
QQQ_Rolling_Future_Return_1m     2.9140      0.005    592.182      0.000       2.904       2.924
==============================================================================
Omnibus:                     1452.365   Durbin-Watson:                   0.296
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            78369.604
Skew:                           0.053   Prob(JB):                         0.00
Kurtosis:                      20.306   Cond. No.                         15.1
==============================================================================

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

                                  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:                 1.566e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:18   Log-Likelihood:                 8264.2
No. Observations:                           6280   AIC:                        -1.652e+04
Df Residuals:                               6278   BIC:                        -1.651e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0059      0.001     -7.038      0.000      -0.008      -0.004
QQQ_Rolling_Future_Return_3m     2.8977      0.007    395.692      0.000       2.883       2.912
==============================================================================
Omnibus:                     1352.162   Durbin-Watson:                   0.102
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15599.280
Skew:                           0.696   Prob(JB):                         0.00
Kurtosis:                      10.594   Cond. No.                         8.95
==============================================================================

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

                                  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:                 8.613e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:19   Log-Likelihood:                 4282.1
No. Observations:                           6280   AIC:                            -8560.
Df Residuals:                               6278   BIC:                            -8547.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0002      0.002     -0.127      0.899      -0.003       0.003
QQQ_Rolling_Future_Return_6m     2.8191      0.010    293.486      0.000       2.800       2.838
==============================================================================
Omnibus:                     1637.681   Durbin-Watson:                   0.057
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8395.060
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.

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_1y   R-squared:                       0.904
Model:                                       OLS   Adj. R-squared:                  0.904
Method:                            Least Squares   F-statistic:                 5.852e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:20   Log-Likelihood:                 404.91
No. Observations:                           6199   AIC:                            -805.8
Df Residuals:                               6197   BIC:                            -792.3
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                            0.0162      0.003      5.033      0.000       0.010       0.023
QQQ_Rolling_Future_Return_1y     2.8920      0.012    241.905      0.000       2.869       2.915
==============================================================================
Omnibus:                     1966.648   Durbin-Watson:                   0.038
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8739.805
Skew:                           1.487   Prob(JB):                         0.00
Kurtosis:                       7.999   Cond. No.                         4.22
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.851
Model:                                       OLS   Adj. R-squared:                  0.851
Method:                            Least Squares   F-statistic:                 3.401e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:22   Log-Likelihood:                -4010.4
No. Observations:                           5977   AIC:                             8025.
Df Residuals:                               5975   BIC:                             8038.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0503      0.008     -6.388      0.000      -0.066      -0.035
QQQ_Rolling_Future_Return_2y     3.2857      0.018    184.405      0.000       3.251       3.321
==============================================================================
Omnibus:                     1728.061   Durbin-Watson:                   0.019
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5277.259
Skew:                           1.488   Prob(JB):                         0.00
Kurtosis:                       6.512   Cond. No.                         3.16
==============================================================================

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

                                  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:                 2.631e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:23   Log-Likelihood:                -5943.8
No. Observations:                           5725   AIC:                         1.189e+04
Df Residuals:                               5723   BIC:                         1.190e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.2891      0.014    -21.164      0.000      -0.316      -0.262
QQQ_Rolling_Future_Return_3y     3.7271      0.023    162.217      0.000       3.682       3.772
==============================================================================
Omnibus:                      846.190   Durbin-Watson:                   0.015
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1556.288
Skew:                           0.941   Prob(JB):                         0.00
Kurtosis:                       4.726   Cond. No.                         3.12
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.789
Model:                                       OLS   Adj. R-squared:                  0.789
Method:                            Least Squares   F-statistic:                 2.045e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:24   Log-Likelihood:                -8114.6
No. Observations:                           5473   AIC:                         1.623e+04
Df Residuals:                               5471   BIC:                         1.625e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.6018      0.024    -25.144      0.000      -0.649      -0.555
QQQ_Rolling_Future_Return_4y     4.3140      0.030    142.997      0.000       4.255       4.373
==============================================================================
Omnibus:                       51.316   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):               40.579
Skew:                           0.131   Prob(JB):                     1.54e-09
Kurtosis:                       2.669   Cond. No.                         3.09
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.750
Model:                                       OLS   Adj. R-squared:                  0.750
Method:                            Least Squares   F-statistic:                 1.574e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:25   Log-Likelihood:                -11327.
No. Observations:                           5238   AIC:                         2.266e+04
Df Residuals:                               5236   BIC:                         2.267e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.4104      0.050    -28.398      0.000      -1.508      -1.313
QQQ_Rolling_Future_Return_5y     5.6451      0.045    125.443      0.000       5.557       5.733
==============================================================================
Omnibus:                      226.987   Durbin-Watson:                   0.009
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              348.438
Skew:                           0.392   Prob(JB):                     2.18e-76
Kurtosis:                       3.990   Cond. No.                         3.11
==============================================================================

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

                                  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:                 5.379e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:26   Log-Likelihood:                 29443.
No. Observations:                           5890   AIC:                        -5.888e+04
Df Residuals:                               5888   BIC:                        -5.887e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -3.034e-05   2.13e-05     -1.426      0.154   -7.21e-05    1.14e-05
QQQ_Rolling_Future_Return_1d     2.9544      0.001   2319.352      0.000       2.952       2.957
==============================================================================
Omnibus:                     8697.849   Durbin-Watson:                   2.575
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         31773160.847
Skew:                          -8.043   Prob(JB):                         0.00
Kurtosis:                     362.454   Cond. No.                         59.9
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:27   Log-Likelihood:                 20303.
No. Observations:                           5890   AIC:                        -4.060e+04
Df Residuals:                               5888   BIC:                        -4.059e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.000     -6.361      0.000      -0.001      -0.000
QQQ_Rolling_Future_Return_1w     2.9525      0.003   1003.867      0.000       2.947       2.958
==============================================================================
Omnibus:                     3288.157   Durbin-Watson:                   0.897
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           369121.603
Skew:                          -1.720   Prob(JB):                         0.00
Kurtosis:                      41.629   Cond. No.                         29.3
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:29   Log-Likelihood:                 13210.
No. Observations:                           5890   AIC:                        -2.642e+04
Df Residuals:                               5888   BIC:                        -2.640e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0025      0.000     -7.377      0.000      -0.003      -0.002
QQQ_Rolling_Future_Return_1m     2.9139      0.005    580.681      0.000       2.904       2.924
==============================================================================
Omnibus:                     1415.259   Durbin-Watson:                   0.310
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            78284.153
Skew:                           0.193   Prob(JB):                         0.00
Kurtosis:                      20.856   Cond. No.                         15.0
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:30   Log-Likelihood:                 7799.9
No. Observations:                           5890   AIC:                        -1.560e+04
Df Residuals:                               5888   BIC:                        -1.558e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0040      0.001     -4.617      0.000      -0.006      -0.002
QQQ_Rolling_Future_Return_3m     2.9067      0.007    388.462      0.000       2.892       2.921
==============================================================================
Omnibus:                     1368.515   Durbin-Watson:                   0.107
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            16450.659
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.

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:31   Log-Likelihood:                 4198.9
No. Observations:                           5890   AIC:                            -8394.
Df Residuals:                               5888   BIC:                            -8380.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0017      0.002     -1.050      0.294      -0.005       0.002
QQQ_Rolling_Future_Return_6m     2.8731      0.010    289.464      0.000       2.854       2.893
==============================================================================
Omnibus:                     1462.300   Durbin-Watson:                   0.063
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8111.743
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.

                                  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:                 6.073e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:32   Log-Likelihood:                 715.33
No. Observations:                           5852   AIC:                            -1427.
Df Residuals:                               5850   BIC:                            -1413.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0009      0.003     -0.283      0.777      -0.007       0.005
QQQ_Rolling_Future_Return_1y     3.0133      0.012    246.430      0.000       2.989       3.037
==============================================================================
Omnibus:                     1763.446   Durbin-Watson:                   0.043
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8317.030
Skew:                           1.385   Prob(JB):                         0.00
Kurtosis:                       8.142   Cond. No.                         4.45
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_2y   R-squared:                       0.865
Model:                                       OLS   Adj. R-squared:                  0.865
Method:                            Least Squares   F-statistic:                 3.714e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:33   Log-Likelihood:                -3517.9
No. Observations:                           5784   AIC:                             7040.
Df Residuals:                               5782   BIC:                             7053.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1117      0.008    -14.133      0.000      -0.127      -0.096
QQQ_Rolling_Future_Return_2y     3.4522      0.018    192.714      0.000       3.417       3.487
==============================================================================
Omnibus:                     1654.899   Durbin-Watson:                   0.020
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5619.104
Skew:                           1.427   Prob(JB):                         0.00
Kurtosis:                       6.895   Cond. No.                         3.36
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.838
Model:                                       OLS   Adj. R-squared:                  0.838
Method:                            Least Squares   F-statistic:                 2.855e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:34   Log-Likelihood:                -5444.7
No. Observations:                           5532   AIC:                         1.089e+04
Df Residuals:                               5530   BIC:                         1.091e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3845      0.014    -27.769      0.000      -0.412      -0.357
QQQ_Rolling_Future_Return_3y     3.9132      0.023    168.966      0.000       3.868       3.959
==============================================================================
Omnibus:                      670.448   Durbin-Watson:                   0.016
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1243.454
Skew:                           0.791   Prob(JB):                    9.71e-271
Kurtosis:                       4.700   Cond. No.                         3.31
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.806
Model:                                       OLS   Adj. R-squared:                  0.806
Method:                            Least Squares   F-statistic:                 2.190e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:36   Log-Likelihood:                -7596.8
No. Observations:                           5280   AIC:                         1.520e+04
Df Residuals:                               5278   BIC:                         1.521e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.7323      0.024    -30.052      0.000      -0.780      -0.685
QQQ_Rolling_Future_Return_4y     4.5109      0.030    147.996      0.000       4.451       4.571
==============================================================================
Omnibus:                       12.319   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.002   Jarque-Bera (JB):               10.050
Skew:                          -0.007   Prob(JB):                      0.00657
Kurtosis:                       2.787   Cond. No.                         3.25
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_5y   R-squared:                       0.759
Model:                                       OLS   Adj. R-squared:                  0.759
Method:                            Least Squares   F-statistic:                 1.625e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:37   Log-Likelihood:                -11052.
No. Observations:                           5158   AIC:                         2.211e+04
Df Residuals:                               5156   BIC:                         2.212e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.6745      0.051    -32.771      0.000      -1.775      -1.574
QQQ_Rolling_Future_Return_5y     5.8696      0.046    127.481      0.000       5.779       5.960
==============================================================================
Omnibus:                      168.503   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              287.431
Skew:                           0.281   Prob(JB):                     3.85e-63
Kurtosis:                       4.010   Cond. No.                         3.27
==============================================================================

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

                                  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:                 4.764e+06
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:38   Log-Likelihood:                 27156.
No. Observations:                           5448   AIC:                        -5.431e+04
Df Residuals:                               5446   BIC:                        -5.429e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                        -1.568e-05   2.24e-05     -0.699      0.485   -5.97e-05    2.83e-05
QQQ_Rolling_Future_Return_1d     2.9531      0.001   2182.718      0.000       2.950       2.956
==============================================================================
Omnibus:                     8179.872   Durbin-Watson:                   2.578
Prob(Omnibus):                  0.000   Jarque-Bera (JB):         30309716.309
Skew:                          -8.324   Prob(JB):                         0.00
Kurtosis:                     368.029   Cond. No.                         60.3
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:39   Log-Likelihood:                 18848.
No. Observations:                           5448   AIC:                        -3.769e+04
Df Residuals:                               5446   BIC:                        -3.768e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0006      0.000     -5.487      0.000      -0.001      -0.000
QQQ_Rolling_Future_Return_1w     2.9495      0.003    967.032      0.000       2.943       2.955
==============================================================================
Omnibus:                     3225.193   Durbin-Watson:                   0.872
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           383535.806
Skew:                          -1.881   Prob(JB):                         0.00
Kurtosis:                      43.932   Cond. No.                         29.6
==============================================================================

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

                                  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:                 3.034e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:40   Log-Likelihood:                 12185.
No. Observations:                           5448   AIC:                        -2.437e+04
Df Residuals:                               5446   BIC:                        -2.435e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0021      0.000     -5.896      0.000      -0.003      -0.001
QQQ_Rolling_Future_Return_1m     2.9063      0.005    550.837      0.000       2.896       2.917
==============================================================================
Omnibus:                     1330.394   Durbin-Watson:                   0.299
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            76723.084
Skew:                           0.205   Prob(JB):                         0.00
Kurtosis:                      21.380   Cond. No.                         15.1
==============================================================================

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

                                  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:                 1.351e+05
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:41   Log-Likelihood:                 7156.4
No. Observations:                           5448   AIC:                        -1.431e+04
Df Residuals:                               5446   BIC:                        -1.430e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0035      0.001     -3.894      0.000      -0.005      -0.002
QQQ_Rolling_Future_Return_3m     2.9141      0.008    367.616      0.000       2.899       2.930
==============================================================================
Omnibus:                     1264.052   Durbin-Watson:                   0.097
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            15799.603
Skew:                           0.751   Prob(JB):                         0.00
Kurtosis:                      11.206   Cond. No.                         9.00
==============================================================================

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

                                  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:                 8.113e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:43   Log-Likelihood:                 4025.3
No. Observations:                           5448   AIC:                            -8047.
Df Residuals:                               5446   BIC:                            -8033.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0043      0.002     -2.530      0.011      -0.008      -0.001
QQQ_Rolling_Future_Return_6m     2.9102      0.010    284.832      0.000       2.890       2.930
==============================================================================
Omnibus:                      970.459   Durbin-Watson:                   0.062
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             4545.928
Skew:                           0.789   Prob(JB):                         0.00
Kurtosis:                       7.188   Cond. No.                         6.55
==============================================================================

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

                                  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:                 5.978e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:44   Log-Likelihood:                 790.32
No. Observations:                           5444   AIC:                            -1577.
Df Residuals:                               5442   BIC:                            -1563.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.0018      0.003     -0.540      0.589      -0.008       0.005
QQQ_Rolling_Future_Return_1y     3.0627      0.013    244.506      0.000       3.038       3.087
==============================================================================
Omnibus:                     1400.332   Durbin-Watson:                   0.045
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             6204.356
Skew:                           1.186   Prob(JB):                         0.00
Kurtosis:                       7.661   Cond. No.                         4.51
==============================================================================

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

                                  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:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:45   Log-Likelihood:                -3103.9
No. Observations:                           5444   AIC:                             6212.
Df Residuals:                               5442   BIC:                             6225.
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.1256      0.008    -15.596      0.000      -0.141      -0.110
QQQ_Rolling_Future_Return_2y     3.5395      0.018    195.774      0.000       3.504       3.575
==============================================================================
Omnibus:                     1478.767   Durbin-Watson:                   0.022
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             5089.206
Skew:                           1.346   Prob(JB):                         0.00
Kurtosis:                       6.898   Cond. No.                         3.45
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_3y   R-squared:                       0.850
Model:                                       OLS   Adj. R-squared:                  0.850
Method:                            Least Squares   F-statistic:                 2.988e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:46   Log-Likelihood:                -5021.0
No. Observations:                           5289   AIC:                         1.005e+04
Df Residuals:                               5287   BIC:                         1.006e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.3933      0.014    -28.260      0.000      -0.421      -0.366
QQQ_Rolling_Future_Return_3y     3.9757      0.023    172.869      0.000       3.931       4.021
==============================================================================
Omnibus:                      578.227   Durbin-Watson:                   0.018
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             1146.794
Skew:                           0.704   Prob(JB):                    9.48e-250
Kurtosis:                       4.795   Cond. No.                         3.36
==============================================================================

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

                                  OLS Regression Results                                 
=========================================================================================
Dep. Variable:     TQQQ_Rolling_Future_Return_4y   R-squared:                       0.820
Model:                                       OLS   Adj. R-squared:                  0.820
Method:                            Least Squares   F-statistic:                 2.303e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:47   Log-Likelihood:                -7126.5
No. Observations:                           5068   AIC:                         1.426e+04
Df Residuals:                               5066   BIC:                         1.427e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -0.7486      0.024    -30.693      0.000      -0.796      -0.701
QQQ_Rolling_Future_Return_4y     4.5873      0.030    151.752      0.000       4.528       4.647
==============================================================================
Omnibus:                       13.154   Durbin-Watson:                   0.011
Prob(Omnibus):                  0.001   Jarque-Bera (JB):               13.239
Skew:                          -0.121   Prob(JB):                      0.00133
Kurtosis:                       2.934   Cond. No.                         3.29
==============================================================================

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

                                  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:                 1.659e+04
Date:                           Mon, 16 Mar 2026   Prob (F-statistic):               0.00
Time:                                   14:27:48   Log-Likelihood:                -10800.
No. Observations:                           5068   AIC:                         2.160e+04
Df Residuals:                               5066   BIC:                         2.162e+04
Df Model:                                      1                                         
Covariance Type:                       nonrobust                                         
================================================================================================
                                   coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------------------------
const                           -1.7612      0.052    -34.112      0.000      -1.862      -1.660
QQQ_Rolling_Future_Return_5y     5.9669      0.046    128.790      0.000       5.876       6.058
==============================================================================
Omnibus:                      145.200   Durbin-Watson:                   0.010
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              276.778
Skew:                           0.210   Prob(JB):                     7.91e-61
Kurtosis:                       4.065   Cond. No.                         3.33
==============================================================================

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)
display(rolling_returns_positive_future_returns)

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_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,
)

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 file. Otherwise, we can download it from Yahoo Finance using the yf_pull_data function.

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"
})

[100%**] 1 of 1 completed

This gives us:

display(spy)

And the plot of the timseries of ajdusted close prices:

plot_timeseries(
    df=spy,
    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_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,
)

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"
})

[100%**] 1 of 1 completed

This gives us:

display(upro)

And the plot of the timseries of ajdusted close prices:

plot_timeseries(
    df=upro,
    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=1,
    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,
)

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 (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)
        )