On the Moments of Ordinary Least Squares and Instrumental Variables Estimators in a General Structural Equation

Abstract

Exact expressions are given for the first two moments of a linear combination of the elements of an instrumental variables estimator for the coefficients of the endogenous variables in a general structural equation. These results generalize previous exact results for equations containing just two or three endogenous variables. In addition, we provide bounds on the moments that can easily be estimated, and generalize some earlier qualita- tive results for these estimators. recently obtained an expression for the joint density of an instrumental variables estimator for the vector of coefficients of the endogenous variables. Prior to this both the exact density and moments were published only for special cases. See Basmann (1), Mariano (20), and Phillips (26) for surveys of these results. When n > 2, however, the exact joint density provides very little information about exact marginal densities and moments for individual coefficients. Phillips (25) has suggested an approximation to the marginal density that seems to work well, but this is so far the only evidence available on marginal densities in the general case. In this paper we derive expressions for the first two moments of a linear combination of the elements of an instrumental variables estimator for the coefficients of the endogenous variables in the general case. These results therefore complement Phillips' (24 and 25) results, and generalize the earlier results of Richardson (27) and Richardson and Wu (28) for the case n = 1. We are also able to generalize some of the other results in these latter two papers. In addition we provide bounds on the exact moments that can easily be estimated in practice, and indicate a procedure for obtaining higher order terms in the asymptotic expansions for the moments given by Nagar (23) and Mikhail (22). Exact marginal densities for individual elements of instrumental variables estima- tors are derived in a separate paper (10). The procedure we follow is, in principle, very simple: starting from the moments of the conditional distribution of the estimator given the right-hand- side endogenous variables, we simply average the conditional moments with respect to the density of the conditioning variates. To accomplish this averaging