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A Generalized Classical Method of Linear Estimation of Coefficients in a Structural Equation

R. L. Basmann

Abstract

The classical method of least-squares estimation of the coefficients $\alpha$ in the (matrix) equation $y$ = $Z /alpha + e$ yields estimators $\Hat{a}$ = $Ay$ = + $Ae$. This method, however, employs only one of a class of transformation matrices, $A$, which yield this result; namely, the special case where $A$ = ($Z \prime{Z}$)-$^1 Z^{\prime}$. As is well known, the consistency of the estimators, $\Hat{a}$, requires that all of the variables whose sample values are represented as elements of the matrix $Z$ be asymptotically uncorrelated with the error terms, $e$. In recent years some rather elaborate methods of obtaining consistent and otherwise optimal estimators of the coefficients $\alpha$ have been developed. In this paper we resent a straightforward generalization of classical linear estimation which leads to estimates of $\alpha$ which possess optimal properties equivalent to those of existing limited-information single-equation estimators, and which is pedagogically simpler and less expensive to apply.