Iterative Estimation of a Set of Linear Regression Equations

Abstract This article develops a method of calculating iterative estimates of the coefficients of a set of linear regression equations. There are p equations such that the explanatory variables are non-stochastic and it is assumed that the random disturbances of at least one pair of equations are correlated. Provided that either different explanatory variables enter the various equations or that if the same explanatory variables enter then the observations are different, it is possible to obtain estimators that are asymptotically more efficient than the simple one-at-a-time least squares estimators. The estimators developed herein are calculated recursively using the Gauss-Seidel method of iteration. It is shown that all of the iterations are asymptotically normal, and that in the limit the iterations have the same asymptotic distribution as the generalized least squares estimator proposed by Zellner. The iterative estimators directly exploit the correlations among the random disturbances. In addition p linear regression equations are defined such that in the ith equation the dependent variable is the residual of the original ith equation and the p − 1 “explanatory” variables are the residuals of all other regressions. It is shown that the regression coefficients of these residuals can be estimated recursively, jointly with the original regression coefficients, or directly from the one-at-a-time least squares residuals. All of these estimators of the regression coefficients of the random disturbances converge in distribution to the same multivariate normal law.