Multiperiod Predictions from Stochastic Difference Equations by Bayesian Methods

Abstract

Given n observations on a system of linear stochastic difference equations with appropriate initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper obtains the predictor of the time series k periods into the future with minimum mean squared error. Completely analytical solution is given for predictions from the first-order univariate system, and, in the general higher-order multivariate case, for k up to 5. econometric equations to produce forecasts were not designed for the purpose of forecasting. In this paper, it is argued that these estimation methods may be inadequate if the resulting estimates are to be used to make ex ante predictions for more than one period ahead, and if the accuracy of the predictions is measured, as it usually is, by the mean squared errors. A formulation of the multiperiod prediction problem is presented. It will then become clear that the same set of parameter estimates cannot be optimal in making predictions for different time periods into the future, when optimality is defined by minimum mean squared errors in small samples. Recently there has been much interest in comparing different econometric models, or different versions of the same econometric model obtained by applying different estimation techniques, in terms of how well they would have forecasted the dependent variables during the sample period, given the true values of the exogenous variables and given the values of the dependent variables lagged one or more periods. It has now become clear that, as judged by ex post forecasting for the sample period, models or techniques that perform better for one-period predictions may do worse for multiperiod predictions. For example, purely auto- regressive models could do better than models based on structural equations in ex post forecasting for one quarter ahead, but were worse in forecasting three or four quarters ahead, as documented in Hickman (4). Klein (9) and Fair (3) have compared multiperiod predictions of the sample data by different estimation techniques applied to the same econometric model. A related, though different, question naturally arises as to whether different parameter estimators should be used to produce ex ante forecasts for different periods into the future. The former topic is one of fitting equations to a set of data. The latter topic is one of statistical decision, and is the subject of this paper.'

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