On Optimal Set Estimation for Partially Identified Binary Choice Models

In this paper we reconsider the notion of optimality in estimation of partially identified models. We illustrate the general problem in the context of a semiparametric binary choice model with discrete covariates as an example of a model which is partially identified as shown in, e.g. Bierens and Hartog (1988). A set estimator for the regression coefficients in the model can be constructed by implementing the Maximum Score procedure proposed by Manski (1975). For many designs this procedure converges to the identified set for these parameters, and so in one sense is optimal. But as shown in Komarova (2013) for other cases the Maximum Score objective function gives an outer region of the identified set. This motivates alternative methods that are optimal in one sense that they converge to the identified region in all designs, and we propose and compare such procedures. One is a Hodges type estimator combining the Maximum Score estimator with existing procedures. A second is a two step estimator using a Maximum Score type objective function in the second step. Lastly we propose a new random set quantile estimator, motivated by definitions introduced in Molchanov (2006). Extensions of these ideas for the cross sectional model to static and dynamic discrete panel data models are also provided.