On the Normalization of Structural Equations: Properties of Direction Estimators

Abstract

In the general structural equation model only the direction of the vector of coefficients of the endogenous variables is determined. The traditional normalization rule defines the coefficients that are of interest but should not be embodied in the estimation procedure: we show that the properties of the traditionally defined ordinary least squares and two stage least squares estimators are distorted by their dependence on the normalization rule. Symmetrically normalized analogues of these estimators are defined and are shown to have essentially similar properties to those of the limited information maximum likelihood estimator. EXACT DISTRIBUTION THEORY for the classical structural equation model in econometrics is notoriously complex, and results for even the simplest cases have seemed too complicated to yield interesting analytical conclusions about the relative merits of different Thus, Anderson and Sawa (1973) and Anderson, Kunimoto, and Sawa (1982), for instance, have resorted to extensive numerical tabulations of the exact densities to extract such information. For the case of an equation with just two endogenous variables these tabulations suggest that, in several respects, the limited information maximum likelihood (LIML) estimator is superior to the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators, among others, and these conclusions are supported by the higher-order asymptotic results in Anderson, Kunimoto, and Morimune (1986) and other work referenced there. The basis for inference in this model is the joint distribution of the included endogenous variables (represented by the reduced form), together with the maintained hypothesis that a submatrix of the reduced form coefficient matrix has rank one less than its column dimension. This condition determines the direction of a vector in Rn + '-where n + 1 is the total number of endogenous variables in the equation-but not its length. Thus, some normalization rule is needed to determine the coefficient vector uniquely, and the distribution theory referred to above has focused on results for the n coefficients remaining when the other is assumed to be unity. In this paper we focus on the estimation of the direction of the (n + 1)- dimensional coefficient vector in an unnormalized equation. Existing distribu- tion results for the coefficients in a normalized equation are easily translated 1An earlier version of this paper bore the title On the interpretation of exact results for structural equation estimators. My thanks to the referees and the Co-Editor for comments on earlier versions of the paper that helped to both clarify the message and broaden the scope of the paper. 1181