Quantile Regression with Interval Data

TL;DR

This paper develops methods for identifying and estimating quantiles and quantile regression parameters when data are set valued, extending traditional concepts and providing computational algorithms and empirical validation.

Contribution

It introduces a novel framework for quantile analysis with set-valued data, including characterization, estimation, inference, and computational algorithms.

Findings

Characterization of quantile sets for set-valued data

Development of estimation and inference methods for various data types

Monte Carlo experiments validate theoretical properties

Abstract

This paper investigates the identification of quantiles and quantile regression parameters when observations are set valued. We define the identification set of quantiles of random sets in a way that extends the definition of quantiles for regular random variables. We then give sharp characterization of this set by extending concepts from random set theory. For quantile regression parameters, we show that the identification set is characterized by a system of conditional moment inequalities. This characterization extends that of parametric quantile regression for regular random variables. Estimation and inference theories are developed for continuous cases, discrete cases, nonparametric conditional quantiles, and parametric quantile regressions. A fast computational algorithm of set linear programming is proposed. Monte Carlo experiments support our theoretical properties.