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, APEδn, is investigated in infinite-order autoregressive (AR(∞)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APEδn is obtained by summing these squared errors from stage nδn, where n is the sample size and 1/n≤δn≤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 APEδn. This expression shows that the prediction performance of APEδn can vary dramatically depending on the choice of δn. Another interesting finding is that when δn approaches 1 at a certain rate, APEδn can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between APEδn 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.