AdvancedQuantitative MethodsPython

Run this module

cd "Quantitative Methods - Regression Analysis"
python "regression_tutorial.py"

Quantitative Methods – Regression Analysis¶

Overview¶

Regression analysis is the statistical "Swiss Army Knife" of quantitative finance. It allows you to quantify relationships between variables, such as how a stock moves relative to the market (Beta) or how factors drive returns.

Key Concepts¶

Linear Regression¶

Equation: $y = \alpha + \beta x + \epsilon$
Beta ($\beta$): Sensitivity of asset to the market
Alpha ($\alpha$): Excess return independent of the market
R-Squared ($R^2$): How well the model explains the data (0 to 1)

Multiple Regression¶

Using multiple independent variables to explain returns.
Example: Fama-French 3-Factor Model (Market, Size, Value).

Diagnostics¶

Residuals: The difference between actual and predicted values. Should be random noise.
t-statistic: Is the coefficient significantly different from zero?
Standard Error: The uncertainty in the estimate.

Key Examples¶

Calculating Beta¶

import numpy as np

# Fit line: Stock Returns = alpha + beta * Market Returns
coeffs = np.polyfit(market_returns, stock_returns, 1)
beta = coeffs[0]
alpha = coeffs[1]

print(f"Beta: {beta:.2f}")

Multiple Regression (Matrix Form)¶

# y = X * beta
# X includes [1, Market, SMB, HML]
X = np.column_stack([np.ones(N), market, smb, hml])
beta = np.linalg.inv(X.T @ X) @ X.T @ y

Files¶

regression_tutorial.py: Interactive tutorial with examples

How to Run¶

pip install numpy scipy
python regression_tutorial.py

Financial Applications¶

1. Beta Calculation (CAPM)¶

Determine how risky a stock is compared to the S&P 500. High beta (>1) means more volatile; low beta (<1) means more stable.

2. Factor Investing¶

Identify which factors (Size, Momentum, Value, Quality) are driving a portfolio's performance using multiple regression.

3. Pairs Trading (Hedge Ratio)¶

Find the optimal hedge ratio between two correlated assets (e.g., Coke vs. Pepsi) to create a market-neutral spread.

4. Predictive Modeling¶

Forecast future returns based on lagged indicators (e.g., dividend yield, interest rates), though this is notoriously difficult!

Best Practices¶

Check Assumptions: Linear regression assumes linear relationship, constant variance (homoscedasticity), and independent errors.
Look at Residuals: If residuals show a pattern, your model is missing something.
Avoid Overfitting: Adding too many variables increases $R^2$ but hurts predictive power. Use Adjusted $R^2$.

Master regression to uncover the hidden drivers of financial markets!

Continue in Quantitative Methods¶

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Factor models explain asset returns as a linear combination of systematic factors plus a stock-specific residual. The Fama-French 3-Factor Model (1992) extended CAPM by adding two well-documented risk premia: the Size premium (SMB) and the Value premium (HML), dramatically improving the explanation of cross-sectional stock returns.

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