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Third-Order Efficiency of the Extended Maximum Likelihood Estimators in a Simultaneous Equation System

Kei Takeuchi
Kimio Morimune

Abstract

We apply the third-order efficient method of estimation to the estimation problem of a system of structural equations in econometrics. The maximum likelihood estimator (hereafter m.l.e.) of structural equations is proved to give uniformly higher probability of concentration about true values than any regular best asymptotically normal estimator, if its asymptotic bias is properly adjusted. For instance, the full-information or limited-information m.l.e give asymptotically uniformly higher probability of concentration than the three-stage or two-stage least-squares estimators, given that these estimators are adjusted to have the same biases. The same result holds for he subsystem m.l.e. We prove the asymptotic completeness of Fuller's modified estimator. Asymptotic expansions of the distributions of the full-information m.l., subsystem m.l., and limited-information m.l. estimators are derived to terms of order O(T^-1). Our general theorem is also applied to the multi-equation seemingly unrelated regression (SUR) model.