Computation of Zellner-Theil's Three Stage Least Squares Estimates

Abstract

In their paper on the three stage least squares method of estimation of a simultaneous equation system, Zellner and Theil [5] make the interesting observation that the large sample efficiency of the estimates of the parameters in the group of over-identified equations is unaffected if the three stage method is applied to this subsystem alone, ignoring all the exactly identified equations. It is shown in this paper that the estimates themselvesnot just their large sample efficiency-are unaffected if one follows the above mentioned simplified procedure. This result is established for the general case of a simultaneous equation system subject to linear homogeneous a priori restrictions, whereas many other known properties of the three stage least squares method have been demonstrated only for the special case of simple restrictions on the structural coefficients. An error in the expression giving the estimates for the exactly identified equations in Zellner-Theil's paper is also corrected. The results of this paper have obvious significance for the problem of developing an efficient computer program for finding the three stage least squares estimates. ZELLNER AND THEIL [5] have proposed a three stage least squares method of estimation of the parameters of a simultaneous equation system as an alternative to the full information maximum likelihood method. The three stage least squares method has been shown to possess a number of attractive properties. Rothenberg and Leenders [2] have shown that when the variance-covariance matrix Z of the disturbance terms is unrestricted, the estimates have the same asymptotic variancecovariance matrix as the full information maximum likelihood method. Under the same condition of unrestricted Z matrix, Sargan [3] has shown that the difference between the estimates by the three stage least squares method and the estimates by the full information maximum likelihood method is of stochastic order 1/T, where T is the sample size. Though the three stage least squares method is much simpler computationally than the full information maximum likelihood method, it still involves the inversion of a moment matrix of a very large order and the difficulties of computing the inverse matrix accurately, especially when it may be near singular due to the presence of intercorrelation among the explanatory