Improving the LIML Estimation with Many Instruments and Persistent Heteroscedasticity

We consider the estimation of coefficients of a structural equation with many instrumental variables in a simultaneous equation system. We propose a class of modifications of the limited information maximum likelihood (MLIML) estimator for improving its asymptotic properties as well as the small sample properties with many instruments and persistent heteroscedasticity. We show that the MLIML estimator improves the LIML estimator and we relate a particular MLIML estimator with the HLIM (or JLIML) estimation. We also give a set of sufficient conditions for an asymptotic optimality when the number of instruments is large with persistent heteroscedasticity. Our method can be extended to the generalized LIML (GLIML) estimation