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October 2005 Order selection for same-realization predictions in autoregressive processes
Ching-Kang Ing, Ching-Zong Wei
Ann. Statist. 33(5): 2423-2474 (October 2005). DOI: 10.1214/009053605000000525


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 efficient (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 same-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.


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Ching-Kang Ing. Ching-Zong Wei. "Order selection for same-realization predictions in autoregressive processes." Ann. Statist. 33 (5) 2423 - 2474, October 2005.


Published: October 2005
First available in Project Euclid: 25 November 2005

zbMATH: 1086.62105
MathSciNet: MR2211091
Digital Object Identifier: 10.1214/009053605000000525

Primary: 60M20
Secondary: 62M10

Keywords: Akaike’s information criterion , Asymptotic efficiency , autoregressive process , order selection , same-realization prediction

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 5 • October 2005
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