Regression Analysis for Risk Management

Simple Linear Regression

Simple linear regression models the relationship between a dependent variable Y and a single independent variable X:

Y = α + βX + ε

Where:

• α = intercept (value of Y when X = 0)
• β = slope (change in Y for a one-unit change in X)
• ε = error term (captures everything the model doesn't explain)

Estimation: Ordinary Least Squares (OLS)

OLS minimizes the sum of squared residuals: Σ(Yᵢ − Ŷᵢ)²

This produces the best linear unbiased estimator (BLUE) when the Gauss-Markov assumptions hold.

Slope formula: β̂ = Σ(Xᵢ − X̄)(Yᵢ − Ȳ) / Σ(Xᵢ − X̄)²

Or equivalently: β̂ = Cov(X, Y) / Var(X)

Intercept: α̂ = Ȳ − β̂X̄

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Multiple Regression

When Y depends on several factors:

Y = α + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε

Each βⱼ represents the partial effect of Xⱼ on Y, holding all other X variables constant.

Interpretation

"A one-unit increase in X₁ is associated with a β₁ change in Y, ceteris paribus."

This "all else equal" interpretation is critical for the FRM:

• In a factor model, βₖ tells you the portfolio's sensitivity to factor k, independent of other factors.
• The intercept α repr

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