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The predictive capability of a modification of Rissanen's accumulatedprediction error (APE) criterion, APE$_{\delta_n}$, is investigated ininfinite-order autoregressive (AR($\infty$)) models. Instead of accumulatingsquares of sequential prediction errors from the beginning, APE$_{\delta_n}$ isobtained by summing these squared errors from stage $n\delta_n$, where $n$ isthe sample size and $1/n\leq \delta_n\leq 1-(1/n)$ may depend on $n$. Undercertain regularity conditions, an asymptotic expression is derived for themean-squared prediction error (MSPE) of an AR predictor with order determinedby APE$_{\delta_n}$. This expression shows that the prediction performance ofAPE$_{\delta_n}$ can vary dramatically depending on the choice of $\delta_n$.Another interesting finding is that when $\delta_n$ approaches 1 at a certainrate, APE$_{\delta_n}$ can achieve asymptotic efficiency in most practicalsituations. An asymptotic equivalence between APE$_{\delta_n}$ and aninformation criterion with a suitable penalty term is also established from theMSPE point of view. This offers new perspectives for understanding theinformation and prediction-based model selection criteria. Finally, we providethe first asymptotic efficiency result for the case when the underlyingAR($\infty$) model is allowed to degenerate to a finite autoregression.

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