Knowledge that Transforms

A. Ronald Gallant, Alberto Holly, Statistical Inference in an Implicit, Nonlinear, Simultaneous Equation Model in the Context of Maximum Likelihood Estimation, Econometrica, Vol. 49, No. 2 (Mar., 1981), p. 554

The exact finite-sample density function of the likelihood ratio (LR) test of overidentifying restrictions in linear equations with any number of included endogenous and exogenous variables is presented, along with its derivation. Properties of the density function and the LR test are presented and discussed. A new approximation for the LR test critical region is proposed and studied. 1. INTRODUCTION AND SUMMARY THE CRITICAL NATURE of identification in simultaneous equation models is well recognized. The identification status of equations within a simultaneous equation system helps determine which estimator to use and dictates some properties of the alternative estimators. It represents a statistical hypothesis that is logically prior to many other standard estimation and testing procedures carried out for model construction and evaluation. The major purposes of this paper are (i) analytical exploration of the likelihood ratio identifiability test statistic for equations with any number of endogenous and exogenous variables and (ii) proposal of a new approximate critical region for testing overidentifying restrictions in such equations. Anderson and Rubin [2] proposed a likelihood ratio test of overidentifying restrictions imposed on the structural parameter space along with their derivation of LIML estimates. Basmann [5] proposed a similar statistic based on the GCL estimation method. McDonald [20] derived the exact finite sample density function for the LR test statistic for an equation containing exactly two endogenous variables. Richardson [26] derived the finite sample density function for the two endogenous variable GCL version of the identifiability test statistic. Both McDonald and Richardson showed that the moments of the exact functions converge to moments of the F distribution as the concentration parameter increases without bound. Each gave necessary conditions for existence of exact moments. McDonald showed that the GCL test statistic will not possess more moments than the LIML statistic in the two-equation case. In further analysis, Kadane [13, 14] showed that both tests are consistent and that both the LR and GCL identifiability test statistics converge to the F distribution as the structural disturbances grow small. Some parallel developments have occurred in multivariate statistical analysis. These are not reviewed here because they are mainly tangential or else are cited at appropriate places in this paper.

Alice Nakamura, Masao Nakamura, A Comparison of the Labor Force Behavior of Married Women in the United States and Canada, with Special Attention to the Impact of Income Taxes, Econometrica, Vol. 49, No. 2 (Mar., 1981), pp. 451-489

[The effects of an improvement in information on the efficiency of risk-bearing are studied under various systems of incomplete markets. With sequential futures markets for uncontingent delivery, the welfare effects are indeterminate in sign, except under special circumstances. In the presence of options markets, however, an improved information structure is almost surely beneficial.]

[A discrete-choice probability model can be estimated from a sample stratified on the choice variable by maximizing the "pseudo-likelihood," a quantity closely related to the log likelihood for a random sample. We investigate the asymptotic properties of the estimator, and show that it is consistent, asymptotically normally distributed, and satisfies a commonly used criterion for asymptotic efficiency.]

[Estimates of parameters in Tobit and other models for limited, truncated and censored dependent variables are not robust against misspecification. A test of the standard assumptions against a general misspecified alternative in the univariate censored normal model is derived and extended to the Tobit regression case. Computational ease and freedom from specification of a specific alternative hypothesis are primary attractions of the test.]

Several procedures are proposed for testing the specification of an econometric model in the presence of one or more other models which purport to explain the same phenomenon.These procedures are shown to be closely related, but not identical, to the non-nested hypothesis tests recently proposed by Pesaran and Deaton [7], and to have similar asymptotic properties..They are remarkably simple both conceptually and computationally, and, unlike earlier techniques, they may be used to test against several alternative models simultaneously.Some empirical results are presented which suggest that the ability of the tests to reject false hypotheses is likely to be rather good in practice.

[The comparative statics of optimization models which have nonlinear constraints are examined. It is shown that most of the standard results of "linear" comparative statics still apply. However, it is also shown that individual substitution and income effects are systematically affected by the nature and degree of nonlinearity of the constraint. A model of quantity/quality trade-offs, previously examined in the literature, is analyzed, and several new results are developed.]