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

• 成果类型: Article
• 署名作者: Ing, Ching-Kang
• 署名单位: Academia Sinica - Taiwan; National Taiwan University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000001550
• 发表日期: 2007
• 页码: 1238-1277
• 关键词: asymptotically efficient selection; stochastic regression-models; least-squares predictors; order selection; identification; inference; bounds

摘要

The predictive capability of a modification of Rissanen's accumulated prediction error (APE) criterion, APES,, is investigated in infinite-order autoregressive (AR(infinity)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APES, is obtained by summing these squared errors from stage n delta(n), where n is the sample size and 1/n <= delta(n) <= 1 - (1/1n) 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 APES,. 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 APES, 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(infinity) model is allowed to degenerate to a finite autoregression.