Non-Standard Rates of Convergence of Criterion-Function-Based Set Estimators for Binary Response Models

Abstract

This paper establishes consistency and non-standard rates of convergence for set estimators based on contour sets of criterion functions for a semiparametric binary response model under a conditional median restriction. The model may be partially identified due to potentially limited-support regressors. A set estimator analogous to the maximum score estimator is essentially cube-root consistent for the identified set when a continuous but possibly bounded regressor is present. Arbitrarily fast convergence occurs when all regressors are discrete. We also establish the validity of a subsampling procedure for constructing confidence sets for the identified set. As a technical contribution, we provide more convenient sufficient conditions on the underlying empirical processes for cube root convergence and a sufficient condition for arbitrarily fast convergence, both of which can be applied to other models. Finally, we carry out a series of Monte Carlo experiments which verify our theoretical findings and shed light on the finite sample performance of the proposed procedures.

Keywords: partial identification, cube-root asymptotics, semiparametric models, limited support regressors, transformation model, binary response model, maximum score estimator

JEL Classification: C13, C14, C25