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TL;DR

A model \( y = X\beta + \epsilon \) where \( \beta \) is estimated by minimizing squared residuals: \( \hat{\beta} = (X^TX)^{-1}X^Ty \). Assumes linear relationship, independent errors, homoscedasticity, and no perfect multicollinearity.

By Valenke Exam Prep Team·Last updated 2026-06-03

Linear Regression

A model \( y = X\beta + \epsilon \) where \( \beta \) is estimated by minimizing squared residuals: \( \hat{\beta} = (X^TX)^{-1}X^Ty \). Assumes linear relationship, independent errors, homoscedasticity, and no perfect multicollinearity.

Why it matters for interviews

Regression is the workhorse of quant finance: factor models (Fama-French), alpha estimation, risk decomposition, and signal construction all use regression. Understanding its assumptions and failure modes is critical.

Definition and Mathematical Foundation

A model \( y = X\beta + \epsilon \) where \( \beta \) is estimated by minimizing squared residuals: \( \hat{\beta} = (X^TX)^{-1}X^Ty \). Assumes linear relationship, independent errors, homoscedasticity, and no perfect multicollinearity.

Application in Quantitative Finance

Regression is the workhorse of quant finance: factor models (Fama-French), alpha estimation, risk decomposition, and signal construction all use regression. Understanding its assumptions and failure modes is critical.

Related Terms

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Frequently Asked Questions

What is the Gauss-Markov theorem?▼
Under OLS assumptions (linearity, exogeneity, homoscedasticity, no autocorrelation), OLS is BLUE: Best Linear Unbiased Estimator. It has the smallest variance among all linear unbiased estimators.
What happens when regression assumptions are violated?▼
Heteroscedasticity: use robust (White) standard errors. Autocorrelation: use Newey-West. Multicollinearity: use ridge regression or PCA. Endogeneity: use instrumental variables.
What is the R-squared and its limitations?▼
\( R^2 \) measures the fraction of variance explained. It always increases with more predictors (use adjusted \( R^2 \)). In finance, \( R^2 \) values of 1-5% for return prediction can be economically significant.