  Assume that observations are generated from an infinite-order autoregressive
[AR($\infty$)] 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.
