Estimation in a Class of Simultaneous Equation Limited Dependent Variable Models

Abstract

Estimation in a class of simultaneous equation limited dependent variable models is considered. The minimum Chi-squared method is used to compare the asymptotic relative efficiency of marginal and new conditional maximum likelihood estimators for this class of models. Efficient minimum Chi-squared estimation procedures are described. A two-step algorithm based on a conditional maximum likelihood estimator provides a natural framework for both computing a linearized and locating the joint maximum likelihood estimator. The unimodality of the simultaneous equation tobit likelihood function is proved and this model is used to illustrate the empirical application of some of the estimators considered in the paper. The relative efficiency of these estimators in the simultaneous equation tobit model is examined in a set of Monte-Carlo experiments. Many applications of microeconomic theory to individual data face the joint problems of censoring and simultaneity. In particular, the dependent variable under investigation may not be continuously observed and some of the conditioning variables representing the outcome of other decisions by the individual may be simultaneously determined. Smith and Blundell (1986) developed an asymptotically efficient test for exogeneity or simultaneity in the simultaneous equation tobit model. As a byproduct, a conditional maximum likelihood estimator was obtained which is consistent under the alternative hypothesis of simultaneity. Nelson and Olsen (1978), Amemiya (1978,1979) and Heckman (1978) consider a number of consistent marginal maximum likelihood estimators based on marginal maximum likelihood estimators of the reduced-form parameters for the probit and tobit models. In contrast, the estimator derived from Smith and Blundell (1986) is based on the corresponding conditional maximum likelihood estimators. This paper is concerned with estimation in a class of simultaneous limited dependent variable regression models which includes the simultaneous probit and tobit models as special cases. Analogous marginal and conditional maximum likelihood estimators to those of Nelson and Olsen (1978), Amemiya (1978, 1979), Heckman (1978) and Smith and Blundell (1986) are derived and other new estimators are suggested. The minimum Chi-squared approach discussed by Ferguson (1958), Malinvaud (1970) and Rothenberg (1973) is used to compare the asymptotic relative efficiency of various estimators considered. A simple two-step algorithm based on the conditional maximum likelihood