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