A New Light from Old Wisdoms : Alternative Estimation Methods of Simultaneous Equations with Possibly Many Instruments

We compare four different estimation methods for a coefficient of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. We prove several theorems on the asymptotic optimality of the LIML estimator when the number of instruments is large, which are new as well as old, and we relate them to the results in some recent studies. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most of interest. We have found that the LIML estimator has good performance when the number of instruments is large, that is, the micro-econometric models with many instruments in the terminology of recent econometric literature.