AIC for many-regressor heteroskedastic regressions

Abstract

The original and corrected Akaike information criteria (AIC) have been routinely used for model selection for ages. The penalty terms in these criteria are tied to the classical normal linear regression, characterized by conditional homoskedasticity and a small number of regressors relative to the sample size. We derive, from the same principles, a general version that takes account of conditional heteroskedasticity and regressor numerosity. The new AICm penalty takes a form of a ratio of certain weighted average error variances. The classical results on asymptotic prediction optimality of AIC are extended to AICm in heteroskedastic possibly misspecified models. Further, the infeasible AICm penalty can be operationalized via unbiased estimation of individual variances. Under suitable conditions, the feasible AICm criterion differs from the infeasible counterpart up to an asymptotically negligible term, and also possesses the property of asymptotic optimality. In simulations, the feasible AICm tends to select models that deliver systematically better out-of-sample predictions than the classical criteria. In the empirical application to an impact of legalized abortion on crime reduction, AICm tends to select a more parsimonious model than non-heteroskedasticity-robust versions of AIC.