Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series

Abstract

The predictive capability of a modification of Rissanen’s accumulated prediction error (APE) criterion, APE_δn, is investigated in infinite-order autoregressive (AR(∞)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APE_δn is obtained by summing these squared errors from stage nδ_n, where n is the sample size and 1/n≤δ_n≤1−(1/n) may depend on n. Under certain regularity conditions, an asymptotic expression is derived for the mean-squared prediction error (MSPE) of an AR predictor with order determined by APE_δn. This expression shows that the prediction performance of APE_δn can vary dramatically depending on the choice of δ_n. Another interesting finding is that when δ_n approaches 1 at a certain rate, APE_δn can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between APE_δn and an information criterion with a suitable penalty term is also established from the MSPE point of view. This offers new perspectives for understanding the information and prediction-based model selection criteria. Finally, we provide the first asymptotic efficiency result for the case when the underlying AR(∞) model is allowed to degenerate to a finite autoregression.