This paper analyzes the linear regression model y = x + with a conditional median assumption Med( j z) = 0 where z is a vector of exogenous random variables. Added complication arise due to the censoring of the outcome y. We treat the censored model as a model with interval-observed outcomes thus obtaining an incomplete model with inequality restrictions on conditional median regressions. This allows us to use the estimator introduced by Manski and Tamer (2000) to analyze the information contained in these inequality restrictions. We give identication conditions in the absence of censoring and introduce a p N-consistent estimator based on the minimum distance method. We then give suÆcient conditions for global identication of with censored y and endogenous x. In the case of interval data on y and endogenous x, we provide a set-consistent estimator that is based on a modied minimum distance method. In the case where we have point identication, we show that the estimator is p N-normal and derive its asymptotic distribution with a feasible asymptotic variance. A Montecarlo analysis illustrates our estimator. We thank Bo Honore for comments and the Econometrics Research Program at Princeton for support.

We discuss the identification and estimation of discrete games of complete information. Following Bresnahan and Reiss (1990, 1991), a discrete game is a generalization of a standard discrete choice model where utility depends on the actions of other players. Using recent algorithms to compute all of the Nash equilibria to a game, we propose simulation-based estimators for static, discrete games. We demonstrate that the model is identified under weak functional form assumptions using exclusion restrictions and an identification at infinity approach. Monte Carlo evidence demonstrates that the estimator can perform well in moderately sized samples. As an application, we study entry decisions by construction contractors to bid on highway projects in California. We find that an equilibrium is more likely to be observed if it maximizes joint profits, has a higher Nash product, uses mixed strategies, and is not Pareto dominated by another equilibrium.

We study inference in structural models with a jump in the conditional density, where location and size of the jump are described by regression curves. Two prominent examples are auction models, where the bid density jumps from zero to a positive value at the lowest cost, and equilibrium job-search models, where the wage density jumps from one positive level to another at the reservation wage. General inference in such models remained a long-standing, unresolved problem, primarily due to nonregularities and computational difficulties caused by discontinuous likelihood functions. This paper develops likelihood-based estimation and inference methods for these models, focusing on optimal (Bayes) and maximum likelihood procedures. We derive convergence rates and distribution theory, and develop Bayes and Wald inference. We show that Bayes estimators and confidence intervals are attractive both theoretically and computationally, and that Bayes confidence intervals, based on posterior quantiles, provide a valid large sample inference method.

Earlier version: June 16, 2004; Revision: June 22, 2006