## The Annals of Statistics

### Order selection for same-realization predictions in autoregressive processes

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 33, Number 5 (2005), 2423-2474.

Dates
First available in Project Euclid: 25 November 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1132936568

Digital Object Identifier
doi:10.1214/009053605000000525

Mathematical Reviews number (MathSciNet)
MR2211091

Zentralblatt MATH identifier
1086.62105

Subjects
Primary: 60M20
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

#### Citation

Ing, Ching-Kang; Wei, Ching-Zong. Order selection for same-realization predictions in autoregressive processes. Ann. Statist. 33 (2005), no. 5, 2423--2474. doi:10.1214/009053605000000525. https://projecteuclid.org/euclid.aos/1132936568

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