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

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

2007-08-17

arxiv.org/abs/0708.2373v1

Annals of Statistics 2007, Vol. 35, No. 3, 1238-1277

Mathematics

Statistics Theory

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

Scientific paper

10.1214/009053606000001550

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.

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