The Maximum Likelihood and the Nonlinear Three-Stage Least Squares Estimator in the General Nonlinear Simultaneous Equation Model

Abstract

The consistency and the asymptotic normality of the maximum likelihood estimator in the general nonlinear simultaneous equation model are proved. It is shown that the proof depends on the assumption of normality, unlike in the linear simultaneous equation model. It is proved that the maximum likelihood estimator is asymptotically more efficient than the nonlinear three-stage least squares estimator if the specification is correct. However, the latter has the advantage of being consistent even when the normality assumption is removed. Hausman's instrumental-variable interpretation of the maximum likelihood estimator is extended to the general nonlinear simultaneous equation model. WE OBTAIN THE ASYMPTOTIC PROPERTIES of the maximum likelihood estimator in the general nonlinear simultaneous equation model and compare them with those of the nonlinear three-stage least squares estimator in this paper. The main results are the following: 1. The proof of the consistency and the symptotic normality of the maximum likelihood estimator in the general nonlinear simultaneous equation model crucially depends on the assumption of normality of the error term, unlike in the linear case. 2. The maximum likelihood estimator is asymptotically more efficient than the nonlinear three-stage least squares estimator if the specification is correct, but it is less robust because the latter is consistent even when the normality assumption is removed. 3. All the third-order derivatives can be asymptotically ignored either in the iterative method for obtaining the maximum likelihood estimator or in the computation of the asymptotic variance-covariance matrix. 4. Hausman's iteration method for the computation of the maximum likelihood estimator in the linear case (see Hausman [3]) is generalized to the nonlinear case. In contrast with the linear case, it does not produce an asymptotically efficient second round estimator even if the initial estimator is consistent, but as in the linear case, it illustrates the similarity and the difference between the maximum likelihood and the nonlinear three-stage least squares estimator.