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, APE$_{\delta_n}$, is investigated in infinite-order autoregressive (AR($\infty$)) models. Instead of accumulating squares of sequential prediction errors from the beginning, 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 APE$_{\delta_n}$. This expression shows that the prediction performance of 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, APE$_{\delta_n}$ can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between 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($\infty$) model is allowed to degenerate to a finite autoregression.