Econometrica: Jul, 1987, Volume 55, Issue 4
Efficient Estimation and Identification of Simultaneous Equation Models with Covariance Restrictions
Jerry A. Hausman, Whitney K. Newey, William E. Taylor
In this paper we consider estimation of simultaneous equations models with covariance restrictions. We first consider FIML estimation and extend Hausman's (1975) instrumental variables interpretation of the FIML estimator to the covariance restrictions case. We show that, in addition to the predetermined variables from the reduced form, FIML also uses estimated residuals as instruments for the equations with which they are uncorrelated. A slight variation on the instrumental variables theme yields a simple, efficient alternative to FIML. Here we augment the original equation system by additional equations that are implied by the covariance restrictions. We show that when these additional equations are linearized around an initial consistent estimator and three-stage least squares is performed on the original equation system together with the linearized equations implied by the covariance restrictions, an asymptotically efficient estimator is obtained. We also present a relatively simple method of obtaining an initial consistent estimator when the covariance restrictions are needed for identification. This estimator also makes use of additional equations that are implied by the covariance restrictions. In the final section of the paper we consider identification from the point of view of the moment restrictions that are implied by instrument-residual orthogonality and the covariance restrictions. We show that the assignment condition of Hausman and Taylor (1983) provides necessary conditions for the identification of the structural parameters.