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.'

[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.]

N this paper, I will generalize the modified Newton method previously applied in Chow (1968) to the computation of full-information maximum likelihood estimates of parameters of a system of linear structural equations to the case of a system of nonlinear structural equations. The success of that method for linear systems 1 has stimulated my present attempt to generalize it for nonlinear systems. The subject of maximum likelihood estimation of nonlinear simultaneous equation systems has been studied by Eisenpress and Greenstadt (1966). There are three main differences between their approach and ours. First, their basic formulation is more general, assuming that all parameters in the system may appear in every equation,2 whereas we assume as the basic setup that there is a distinct set of parameters belonging to each equation. Second, partly because of the first, we are able to obtain simpler and more explicit expressions for the derivatives of likelihood function required in the calculations. Third, and also partly because of the first, we can conveniently deal with the important problem of linear restrictions on the parameters in the same equation or in different equations. A fourth feature of this paper, and a feature which has partly motivated it, is the contrast of the linear with the nonlinear case. As it will be shown, there are many similarities in the computations of both. This demonstration can enhance our understanding of the nature of the estimation equations. Two additional features of this paper are the treatments of identities in the system and of residuals which may follow an autoregressive scheme. We will derive in section II the estimation equations for nonlinear systems, under the assumptions that each structural equation contains a distinct set of parameters, that the parameters are not subject to any linear restrictions, and that the (additive) residuals are serially uncorrelated. Section III treats the special case when some equations are linear, and contrasts this case with the nonlinear case. Section IV deals with identities and linear restrictions on the parameters. Section V is concerned with the problem of autoregressive residuals.