ASYMPTOTICALLY EFFICIENT AUTOREGRESSIVE MODEL SELECTION FOR MULTISTEP PREDICTION

R. J. BHANSALI

Department of Statistics and Computational Mathematics, University of Liverpool, Victoria Building, Brownlow Hill, P.O. Box 147, Liverpool L69 3BX, U.K.

(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, k̃_{h}, 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 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.

Key words and phrases: AIC, FPE, order determination, time series.