Non‐standard rates of convergence of criterion‐function‐based set estimators for binary response models

Abstract

Summary This paper establishes consistency and non-standard rates of convergence for set estimators based on contour sets of criterion functions for a semi-parametric binary response model under a conditional median restriction. The model can be partially identified due to potentially limited-support regressors and an unknown distribution of errors. 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.