An Instrumental Variable Approach to Full Information Estimators for Linear and Certain Nonlinear Econometric Models

Abstract

FIML is shown to be an instrumental variables estimator where the instruments embody all the over-identifying a priori restrictions. FIML is compared to the two alternative estimators 3SLS and full information instrumental variables. 3SLS differs from FIML in not using all a priori restrictions in forming the instruments. The full information instrumental variables estimator when iterated to convergence yields the FIML estimate. For the case of nonlinearity in the parameters a nonlinear 3SLS and a nonlinear full information instrumental variables estimator are proposed. Both estimators are asymptotically efficient. THIS PAPER UNDERTAKES an investigation of asymptotically efficient estimators for linear and nonlinear simultaneous equation econometric models. By using an instrumental variable approach the equivalence of previously proposed linear estimators to full information maximum likelihood (FIML) follows in a straightforward manner, and a class of new estimators which includes a nonlinear three stage least squares estimator (NL3SLS) and nonlinear full information instrumental variables estimator are proposed and shown to be asymptotically equivalent to FIML. First, an instrumental variable interpretation of FIML is developed by investigating the first order conditions for the maximum of the likelihood function without first concentrating the likelihood function. The essential difference between 3SLS and FIML then becomes evident. The difference between the two estimators is first that FIML uses all over-identifying restrictions in forming the instruments while 3SLS ignores some of these restrictions. Also, FIML uses an estimate of the covariance matrix in forming the instruments which is consistent in the sample with the parameter estimates. Thus the instruments used by FIML are mutually consistent with the parameter estimates in the given sample, while for other estimators the instruments are consistent with the parameter estimates only asymptotically. While this difference in forming the instruments is of no importance asymptotically as is known by the earlier results of Sargan [10] and Rothenberg and Leenders [9], in finite samples there seems to be no reason for not using all known prior information. The use of the a priori restrictions gives a more useful criterion than Dhrymes' [3] recent interpretations of a difference in purging the endogenous variables since all other proposed estimators can be shown to be equivalent by simply proving asymptotic equivalence of the instruments used to those instruments used by the FIML estimator. Then using the instrumental variable interpretation, a relation between FIML and the class of estimators recently proposed by Dhrymes [2], Lyttkens [5, 6], and Brundy and Jorgenson [1] is established. The full information instrumental