Multiple Linear Regression

Many predictors, one response — and the interpretations get subtle.

Multiple linear regression generalizes simple regression to many predictors:

y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_p x_p + \epsilon

Each $\hat{\beta}_j$ is the effect of $x_j$ on $y$ holding the other predictors fixed. This conditional interpretation is what makes multiple regression powerful, and it's also what gets confused most often.

Three traps to know:

Multicollinearity. If predictors are highly correlated, individual coefficient variances explode and standard errors balloon. The model still predicts well, but you can't reliably interpret individual coefficients.Omitted variable bias. Leaving out a relevant predictor that correlates with included ones biases the included coefficients.Interaction effects. Adding $\beta_{12} x_1 x_2$ lets the effect of $x_1$ depend on the value of $x_2$ — important whenever effects are non-additive.

In quant trading, multivariate regressions on factor returns are the foundation of factor models. The coefficients are the loadings — how much your portfolio moves with the market, with size, with value, and so on.