Asymptotically efficient autoregressive model selection for multistep prediction

A direct method for multistep prediction of a stationary time series involves fitting, by linear regression, a different autoregression for each lead time, h, and to select the order to be fitted, (k) over tilde(h), from the data. By contrast, a more usual 'plug-in' method involves the least-squares fitting of an initial Ic-th order autoregression, with k itself selected by an order selection criterion. A bound for the mean squared error of prediction of the direct method is derived and employed for defining an asymptotically efficient order selection for h-step prediction, h greater than or equal to 1; the S-h(k) criterion of Shibata (1980) is asymptotically efficient according to this definition. A bound for the mean squared error of prediction of the plug-in method is also derived and used for a comparison of these two alternative methods of multistep prediction. Examples illustrating the results are given.