Random Set Quantile Estimation of Partially Identified Discrete Response Models

2026-06-01

AI Summary

Discrete covariates often render parameters in discrete choice models only partially identified, leading to severe ill-posedness in classical estimators. This paper proposes a Random Set Quantile (RSQ) estimation approach that, for the first time, reveals the convergence of classical estimators to a random set. By extracting the τ-quantile of this limiting random set over the interval (1/2, 1), the method constructs a novel estimation framework that is uniformly consistent and locally robust across the entire parameter space. Integrating random set theory, quantile estimation, and m-out-of-n bootstrap techniques, the RSQ estimator effectively addresses partial identification induced by Brexit-related covariates in UK 2019 general election data, demonstrating both empirical validity and robustness.

Abstract

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 $τ$-quantile of the classical estimator for $τ\in (1/2,1)$. 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.

Problem

• partial identification
• discrete response models
• random set
• quantile estimation
• semiparametric estimation

Innovation

• Random Set Quantile
• Partial Identification
• Discrete Response Models
• Semiparametric Estimation
• m-out-of-n Bootstrap