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

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 0 < δn < 1 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 performances 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. It offers a new perspective for comparing the informationand prediction-based model selection criteria in AR(∞) models. Finally, we provide the first asymptotic efficiency result for the case when the underlying AR(∞) model is allowed to degenerate to a finite autoregression.