The Moment Matrix of the Two-Stage Least-Squares Estimator of Coefficients in Different Equations of a Complete System of Simultaneous Equations

Abstract

IN A SYSTEM of simultaneous equations one may estimate the coefficients by the two-stage least-squares procedure proposed by Theil [4] or by any other alternative method. While studying the significance of a linear combination (for example, the sum or difference) of coefficients in different equations, one requires the knowledge of variances and covariances of the coefficient estimators. Theil [4] gave the asymptotic covariance matrix of the two-stage least-squares estimator of coefficients in two different equations. The asymptotic covariance matrix of the two-stage least-squares estimator of coefficients in one single equation can, of course, be treated as a particular case. Nagar [2] worked out the bias to order 1/T, T being the number of observations, and the moment matrix to order 1/T2, of the two-stage least-squares estimator of coefficients in a single equation. The object of the present article is to derive the moment matrix, to order 1/T2, of the two-stage least-squares estimator of coefficients in two different equations of the system. The moment matrix (to order 1/T2) of the two-stage least-squares estimators of coefficients in a single equation will then be a particular case of the above result. The following section of this article gives the main theorem. The proof of the theorem is given in Section 3.