The Error of Prediction for a Simultaneous Equation Model

One of the most important functions of a simultaneous equation model is prediction the values of endogenous variables given the values of the predetermined variables and a lot of work has been done to estimate the accuracy of such predictions. Hooper and Zellner (1961) obtained the covariance matrix of the prediction error for unrestricted reduced form and Goldberger, Nagar and Odeh (1961) derived one for restricted reduced form. Properties of predictions for partially restricted reduced form have been analyzed by Amemiya (1966), Kakwani and Court (1972) and Nagar and Sahay (1978). The comparison of these estimators in the context of prediction has been carried on by Dhrymes (1973) and Park (1982). However all these derivations are made for reduced forms of correctly specified linear simultaneous equation models and they still remain unknown for the under and the over specified models. The purpose of this paper is to derive the matrices of the mean squared prediction error for both the underfitted and the overfitted models of unrestricted reduced form of a linear simultaneous equation system. The paper is organized as follows: Section 2 presents the basic model and its assumptions. Sections 3 and 4 derive the matrices of the mean squared prediction error for the underfitted and the overfitted models of unrestricted reduced form respectively. Section 5 gives the conclusions. An appendix contains the proofs of these derivations.