Misspecification and Conditional Maximum Likelihood Estimation

Recently White (1982) studied the properties of Maximum Likelihood estimation of possibly misspecified models. The present paper extends Andersen (1970) results on Conditional Maximum Likelihood estimators (CMLE) to such a situation. In particular, the asymptotic properties of CMLE's are derived under correct and incorrect specification of the conditional model. Robustness of conditional inferences and estimation with respect to misspecification of the model for the conditioning variables is emphasized. Conditions for asymptotic efficiency of CMLE's are obtained, and specification tests a la Hausman (1978) and White (1982) are derived. Examples are also given to illustrate the use of CMLE's properties. These examples include the simple linear model, the multinomial logit model, the simple Tobit model, and the multivariate logit model

Cramer-Rao Bounds for Misspecified Models

In this paper, we derive some lower bounds of the Cramer-Rao type for the covariance matrix of any unbiased estimator of the pseudo-true parameters in a parametric model that may be misspecified. We obtain some lower bounds when the true distribution belongs either to a parametric model that may differ from the specified parametric model or to the class of all distributions with respect to which the model is regular. As an illustration, we apply our results to the normal linear regression model. In particular, we extend the Gauss-Markov Theorem by showing that the OLS estimator has minimum variance in the entire class of unbiased estimators of the pseudo-true parameters when the mean and the distribution of the errors are both misspecified

Probability Feedback in a Recursive System of Probability Models

This paper presents a general model for qualitative endogenous variables that is defined by a recursive system of conditional probability models in which the probabilities of some outcomes may depend on the probabilities of posterior outcomes. The model is related to, but conceptually different from C. D. Mallar's (1977) simultaneous probability model. It has as special cases the multivariate logit model (M. Nerlove and S. J. Press (1973, 1976)) and the constrained nested logit model (D. McFadden (1981)). The model can also be used to analyze outcomes of some game situations. Two examples are in particular considered: a game against Nature and a Stackelberg game under uncertainty. Identification of the structural parameters in the first example is seen to be related to the classical problem of stochastic revealed preference as studied by M. K. Richter and L. Shapiro (1978)

Generalized Inverses and Asymptotic Properties of Wald Tests

We consider Wald tests based on consistent estimators of g-inverses of the asymptotic covariance matrix ∑ of a statistic that is n^1/2-asymptotically normal distributed under the null hypothesis. Under the null hypothesis and under any sequence of local alternatives in the column space of ∑, these tests are asymptotically equivalent for any choice of g-inverses. For sequences of local alternatives not in the column space of ∑ and for a suitable choice of g- inverse, the asymptotic power of the corresponding Wald test can be made equal to zero or arbitrarily large. In particular, the test based on a consistent estimator of the Moore-Penrose inverse of ∑ has zero asymptotic power against sequences of local alternatives in the orthogonal complement to the column space of ∑

Selecting the Best Linear Regression Model: A Classical Approach

In this paper, we apply the model selection approach based on Likelihood Ratio (LR) tests developed in Vuong (1985) to the problem of choosing between two normal linear regression models which are not nested in each other. First we compare our model selection procedure to other model selection criteria. Then we explicitly derive the procedure when the competing linear models are non-nested and neither one is correctly specified. Some simplifications are seen to arise when both models are contained in a larger correctly specified linear regression model, or when at least one competing linear model is correctly specified. A comparison of our model selection tests and previous non-nested hypothesis tests concludes the paper

Modèles d’équations simultanées pour variables endogènes fictives : une formulation par la théorie des jeux avec application à la participation au marché du travail

Les auteurs proposent une méthode inspirée par la théorie des jeux pour formuler des modèles d’équations simultanées et les appliquer à l’étude de la décision de deux personnes mariées de participer ou pas au marché du travail. Leur démarche se distingue par le fait que le modèle simultané utilise le résultat d’un jeu entre deux joueurs pour obtenir le comportement d’optimisation. Le concept d’équilibre retenu est celui de Nash. Par ailleurs, les auteurs démontrent que les conditions de cohérence logique, sont implicites dans les modèles d’équations simultanées avec glissement structurel, interdiraient en fait la simultanéité dans la modélisation du problème qu’ils considèrent. Par conséquent, leur modèle ne suppose aucune condition de cohérence logique dans les paramètres.A game theoretic approach for formulating simultaneous equations models for dummy endogenous variables is proposed and applied to a study of husband/wife labor force participation. A distinctive feature of our approach is that the simultaneous model is derived from optimizing behavior as an outcome of a game between two players. The equilibrium concept used is that of Nash. In addition, we show that the logical consistency conditions implied by usual simultaneous equation models with structural shift actually rules out simultaneity for the problem we consider; in our model, no logical consistency conditions are implied on the parameters

A Note on the Independence of Irrelevant Alternatives in Probabilistic Choice Models

The purpose of this note is to show that there is no necessary relationship between the independence of irrelevant alternatives (IIA) property and stochastic independence of the errors in probabilistic choice models

Generalized Inverses and Asymptotic Properties of Wald Tests

We consider Wald tests based on consistent estimators of g-inverses of the asymptotic covariance matrix ∑ of a statistic that is n^1/2-asymptotically normal distributed under the null hypothesis. Under the null hypothesis and under any sequence of local alternatives in the column space of ∑, these tests are asymptotically equivalent for any choice of g-inverses. For sequences of local alternatives not in the column space of ∑ and for a suitable choice of g- inverse, the asymptotic power of the corresponding Wald test can be made equal to zero or arbitrarily large. In particular, the test based on a consistent estimator of the Moore-Penrose inverse of ∑ has zero asymptotic power against sequences of local alternatives in the orthogonal complement to the column space of ∑

Misspecification and Conditional Maximum Likelihood Estimation

Recently White (1982) studied the properties of Maximum Likelihood estimation of possibly misspecified models. The present paper extends Andersen (1970) results on Conditional Maximum Likelihood estimators (CMLE) to such a situation. In particular, the asymptotic properties of CMLE's are derived under correct and incorrect specification of the conditional model. Robustness of conditional inferences and estimation with respect to misspecification of the model for the conditioning variables is emphasized. Conditions for asymptotic efficiency of CMLE's are obtained, and specification tests a la Hausman (1978) and White (1982) are derived. Examples are also given to illustrate the use of CMLE's properties. These examples include the simple linear model, the multinomial logit model, the simple Tobit model, and the multivariate logit model