Multiple Linear Regression: Model Building for Actuaries
Learn multiple linear regression, interpretation, and model building for Exam SRM.
The Multiple Regression Model
Multiple linear regression extends the simple model to p predictors: Y = beta_0 + beta_1*X1 + beta_2*X2 + ... + beta_p*Xp + epsilon. In matrix notation, Y = X*beta + epsilon, and the OLS solution is beta_hat = (X'X)^(-1)*X'Y. The interpretation changes: beta_j represents the expected change in Y for a one-unit increase in Xj, holding all other predictors constant.
The "holding constant" interpretation is crucial and distinguishes multiple regression from running separate simple regressions. Coefficients can change sign or magnitude when additional predictors are added.
R-Squared and Adjusted R-Squared
R^2 = 1 - SSE/SST always increases (or stays the same) when predictors are added, even if they are irrelevant. This makes R^2 unreliable for model selection. Adjusted R^2 = 1 - (SSE/(n-p-1))/(SST/(n-1)) penalizes for the number of predictors and can decrease when an unhelpful predictor is added. Use adjusted R^2 for comparing models with different numbers of predictors.
F-Test for Overall Significance
The overall F-test tests H0: beta_1 = beta_2 = ... = beta_p = 0 (no predictors are useful). F = (SSR/p) / (SSE/(n-p-1)) = MSR/MSE follows an F-distribution with (p, n-p-1) degrees of freedom under H0. A partial F-test compares nested models: F = ((SSE_reduced - SSE_full) / q) / MSE_full, where q is the number of additional parameters in the full model. This tests whether the extra predictors significantly improve the model. Both tests are important for Exam SRM.