Random Set Quantile Estimation of Partially Identified Discrete Response Models

Semiparametric discrete choice models are widely applied in economics, yet a fundamental tension arises when covariates are discrete as regression coefficients that are point identified under continuous regressors may become only partially identified. We show that this is not merely an identification problem but creates serious estimation pathologies. Classical estimators, including the maximum score estimator of Manski (1975), not only have population maximizers that are outer regions of the identified set (Komarova (2013)) but also converge to a random set drawn from a finite collection of deterministic regions that partition that outer region. To resolve this failure, we introduce the Random Set Quantile (RSQ) estimator which extracts the $\tau$-quantile of the classical estimator for $\tau \in (1/2,1)$. We prove this result for a class of widely used models, which includes binary/multinomial choice and discrete outcome panel data models. This construction is consistent and locally robust across the full parameter space, including precisely those configurations where classical estimators break down. A feasible implementation based on the $m$-out-of-$n$ bootstrap inherits both properties. We apply the methodology to the 2019 UK General Election, where the discrete support of Brexit-related covariates generates the partial identification our theory analyzes.