(Received March 1, 1994; revised June 5, 1995)
Abstract. 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, ~kh, from the data. By contrast, a more usual `plug-in' method involves the least-squares fitting of an initial k-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 > 1; the Sh(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.
Key words and phrases: AIC, FPE, order determination, time series.