Likelihood Estimation and Inference in a Class of Nonregular Econometric Models

Abstract

In this paper we study estimation and inference in structural models with a jump in the conditional density, where the location and size of the jump are described by regression lines. Two prominent examples are auction models, where the density jumps from zero to a positive value, and the equilibrium job search model, where the density jumps from one level to another, inducing kinks in the cumulative distribution function. An early model of this kind was introduced by Aigner, Amemiya, and Poirier (19776), but the estimation and inference in such models remained an unresolved problem, with the important exception of the specific cases studied by Donald and Paarsch (1993a) and the univariate case in Ibragimov and Has'minskii (1981a). The main difficulty is the statistical non-regularity of the problem caused by discontinuities in the likelihood function. This difficulty also makes the problem computationally challenging. This paper develops estimation and inference theory and methods for such models based on likelihood procedures, focusing on the optimal (Bayes) procedures, including the MLEs. We obtain results on convergence rates and distribution theory, and develop Wald and Bayes type inference and confidence intervals. The Bayes procedures are attractive both theoretically and computationally. The Bayes confidence intervals, based on the posterior quantiles, are shown to provide a valid large sample inference method with good small sample properties. This inference result is of independent practical and theoretical interest due to the highly non-regular nature of the likelihood in these models, in which the maximum likelihood statistic or any finite dimensional statistic is not asymptotically sufficient.