A family of tests of significance is developed for coefficients in a single equation of a simultaneous system., Different members of this family are distinguished by the k-class estimator on which they are based, and on the alternative hypothesis against which they test. The size of the test is found when the disturbances are small, and the test is shown to be consistent if plim k = 1 or k = λ, the limited information maximum likelihood value.

A new approach to the choice of econometric estimators, called small-sigma asymptotics, is introduced and applied to the choice of k-class estimators of the parameters of a single equation in a system of linear simultaneous stochastic equations. I find that when the degree of overidentification is no more than six, the two stage least squares estimator uniformly dominates the limited information maximum likelihood estimator in a certain sense. The small sigma method can be used on many problems in statistics and econometrics. THE STUDY OF simultaneous equation econometric models has led to many alternative estimators to ordinary least squares: single equation limited information maximum likelihood, and two stage least squares, for example. The behavior of these estimators has been difficult to describe, however, and it has been difficult to choose among these estimators. The work described in this paper explores this problem for the case in which lagged dependent variables are not permitted. To be most useful for normative purposes, a description must be detailed enough to give a good approximation and expose differences between estimators, and yet be simple enough to strengthen intuition and yield easily described comparisons. Since detail and simplicity are in conflict, approaches may differ in this respect. This paper introduces a new approach, based on asymptotic series in a scalar multiple, a, of the variance of the disturbance in the model. As a -+ 0 the regression function is an increasingly good description of the random variables generated. Intuitively this is suggested by Gauss' Theory of Errors the errors were never intended to be so as to swamp the regression function. One important approach used in the past is sample asymptotic theory. This reveals a persistent bias in ordinary least squares, and a sample asymptotic equivalence between two stage least squares and single equation limited information maximum likelihood. Additionally, Nagar [13] found the 11T term in the sample asymptotic bias and the 1/T and I/T2 terms of the moment matrix of two stage least squares. Economists have been uneasy, however, about application of sample theory to samples which may not be large in the relevant sense. Additionally sample asymptotic results often depend on an assumption about the asymptotic behavior of the moment matrix of exogenous variables which is difficult to justify.

IN THEIR ARTICLE [5], Liu and Breen propose a new estimator of the large-sample asymptotic covariance matrix for the limited information maximum likelihood estimator in simultaneous equations, and express surprise that their estimator is different from the estimator proposed by Chernoff and Divinsky [1]. Additionally, they question the interpretation of a statistic used in the past to test over-identifying restrictions.