Multicollinearity: Detection and Remediation for Actuaries

What Is Multicollinearity?

Multicollinearity occurs when two or more predictors are highly correlated with each other. This does not violate the regression assumptions or bias the coefficient estimates, but it inflates their standard errors, making individual coefficients unreliable. The overall model fit (R^2, F-test) may be fine even when individual t-tests are all insignificant. This paradox is a hallmark of multicollinearity.

Perfect multicollinearity (one predictor is an exact linear combination of others) makes (X'X) singular, so the OLS solution does not exist. Near-perfect multicollinearity allows estimation but produces unstable results.

Detection

The Variance Inflation Factor (VIF) for predictor j is VIF_j = 1 / (1 - R_j^2), where R_j^2 is the R-squared from regressing Xj on all other predictors. VIF > 5 or 10 suggests problematic multicollinearity. The condition number of X'X also measures multicollinearity: values above 30 indicate concern. Pairwise correlations above 0.8 or 0.9 are another indicator, though they miss multicollinearity involving three or more variables.

Remediation

Approaches include: removing one of the correlated predictors, combining correlated predictors into a single variable (e.g., using their sum or ratio), applying principal component regression, or using regularization (ridge regression, which shrinks coefficients and stabilizes them). On Exam SRM, the most commonly tested remediation is removing a predictor or using ridge/LASSO regularization. Remember that multicollinearity is mainly a problem for interpretation and inference; if the goal is purely prediction and the multicollinearity pattern is stable, it may not need correction.