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ASYMPTOTICALLY EFFICIENT AUTOREGRESSIVE MODEL

SELECTION FOR MULTISTEP PREDICTION

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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,
^{~}*k*_{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.

**Source**
( TeX ,
DVI ,
PS )