• DOI:10.1214/009053605000000525
• Corpus ID: 18814602

Order selection for same-realization predictions in autoregressive processes

@article{Ing2005OrderSF,
  title={Order selection for same-realization predictions in autoregressive processes},
  author={Ching-Kang Ing and Ching-Zong Wei},
  journal={Annals of Statistics},
  year={2005},
  volume={33},
  pages={2423-2474}
}

• C. Ing, Ching-Zong Wei
• Published 1 October 2005
• Computer Science
• Annals of Statistics

Assume that observations are generated from an infinite-order autoregressive [AR(∞)] process. Shibata [Ann. Statist. 8 (1980) 147-164] considered the problem of choosing a finite-order AR model, allowing the order to become infinite as the number of observations does in order to obtain a better approximation. He showed that, for the purpose of predicting the future of an independent replicate, Akaike's information criterion (AIC) and its variants are asymptotically efficient. Although Shibata's concept of asymptotic efficiency has been widely accepted in the literature, it is not a natural property for time series analysis. This is because when new observations of a time series become available, they are not independent of the previous data. To overcome this difficulty, in this paper we focus on order selection for forecasting the future of an observed time series, referred to as same-realization prediction. We present the first theoretical verification that AIC and its variants are still asymptotically elficient (in the sense defined in Section 4) for same-realization predictions. To obtain this result, a technical condition, easily met in common practice, is introduced to simplify the complicated dependent structures among the selected orders, estimated parameters and future observations. In addition, a simulation study is conducted to illustrate the practical implications of AIC. This study shows that AIC also yields a satisfactory saute-realization prediction in finite samples. On the other hand, a limitation of AIC in same-realization settings is pointed out. It is interesting to note that this limitation of AIC does not exist for corresponding independent cases.