Order Selection for the Same-realization Prediction in Autoregressive Processes

PREDICTION IN AUTOREGRESSIVE PROCESSES C. K. ING AND C. Z. WEI National Taipei University and Academia Sinica Abstract Assume observations are generated from an infinite-order autoregressive (AR) process. Shibata (1980) 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, AIC and its variants are asymptotically efficient. Although Shibata’s concept of asymptotic efficiency has been wildly 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, our emphasis of order selection is put on the same-realization prediction, which aims at the future of the observed time series. We present the first theoretical justification that AIC and its variants still possess the asymptotic efficiency for the same-realization prediction. To prove this result, a moment bound for the norm of the inverse sample covariance matrix with an increasing dimension is established. In addition, a new condition is also introduced to handle the messy dependence structures among the model selector, estimated parameters and future observation.