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, 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 $1/n\le {\delta }_n\le 1-(1/n)$ 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.

Published at http://dx.doi.org/10.1214/009053606000001550 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Order selection for same-realization predictions in autoregressive processes
• Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series

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