Approximate Distributions of k-Class Estimators when the Degree of Overidentifiability is Large Compared with the Sample Size

In the estimation of structural coefficients it is well-known that both two-stage least squares (TSLS) and limited information maximum likelihood (LIML) estimators are consistent and asymptotically efficient, and that the exact mean of the LIML estimator does not exist. Then the TSLS estimator, which is computationally simpler, has appeared a proper choice to empirical researchers. In this article asymptotic properties of the k-class and related estimators are sorted out according to the ratio between the total number of exogenous variables and the number of observations. It is found that the TSLS distribution deviates far from its traditional asymptotic distribution; the LIML distribution stays stable about its traditional asymptotic distribution. The LIML estimator now seems more attractive than the TSLS estimator except for the fact that its exact moments do not exist. A modified estimator is proposed which is asymptotically better than the LIML estimator and whose exact moments exist.