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

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.