On the Treatment of Autocorrelated Errors in the Multiperiod Prediction of Dynamic Simultaneous Equation Models

Abstract

Estimation of parametric multiple time series models has been a major topic in recent work in statistics and econometrics (e.g., Hannan [1970], Wilson [1973], Dhrymes and Erlat [1974], and Hatanaka [1976]). However, relatively little has been reported on their prediction property, particularly when the involves autocorrelated errors. For the with uncorrelated errors, Goldberger, Nagar and Odeh [1962] and Dhrymes [1973] have obtained the asymptotic distribution of the reduced form coefficient estimates derived from the structural form estimates. It obviously serves as the asymptotic distribution of the one period ahead prediction of the model. Schmidt [1974] recently derived the asymptotic distribution of multiperiod ahead predictions for such a model, i.e., simultaneous equation autoregressive (or alternatively called dynamic model in econometrics) with exogenous variables (ARX) (see also Brissimis and Gill [1978]). When the disturbances of the models are autocorrelated, it is known that the derivation of the simplified prediction scheme, not to mention its asymptotic distribution, becomes complicated even for the single equation model. Bloomfield [1972] and Yamamoto [1978] derived the asymptotic mean square error of one period and multiperiod predictions for the single equation autoregressive moving average (ARMA) models, respectively. In this paper, first we derive the optimal prediction scheme for multiperiod prediction of a simultaneous equation autoregressive with exogenous variables, whose disturbances obey either autoregressive or moveing average process. The complication due to the error autocorrelation is handled by the introduction of the backward representation of the model, and the optimal predictor is given by a relatively simple formula with matrix notations. Secondly, for the unknown parameter case, we derive the asymptotic distribution of the optimal prediction scheme with the consistent estimates of the parameters. Our results are quite general, and we show, with a few examples, that they are easily modified to various single and simultaneous equation models of simpler specifications. The scope of this paper is as follows. Section 2 presents two types of representation. The first is the state variable representation suggested by