Estimation and Inference for the $τ$-Quantile of Individual Heterogeneous Coefficient

2026-05-03

AI Summary

This study addresses the estimation and inference of τ-quantiles of heterogeneous slope coefficients in panel data, moving beyond the conventional focus on outcome heterogeneity. It proposes a two-step quantile regression framework to characterize the quantile features of the cross-sectional distribution of slope coefficients and establishes asymptotic theory under both random and fixed designs. The method relaxes conventional sample growth conditions, making it suitable for large-N settings and accommodating both √N and √(N√T) convergence rates. Two bootstrap procedures are developed to facilitate valid inference. Monte Carlo simulations and an empirical application to mutual fund data demonstrate the method’s effectiveness and reveal substantial heterogeneity in slope coefficients across different quantiles.

Abstract

This paper proposes estimation and inference procedures for the quantiles of individual heterogeneous slope coefficients within panel data. We develop a two-step quantile estimation framework for analyzing heterogeneity in individual coefficients. Unlike conventional panel quantile regression, which focuses on outcome heterogeneity, our approach targets the $τ$-quantile of the cross-sectional distribution of individual-specific slopes. We establish asymptotic theory under both stochastic and deterministic designs, with convergence rates $\sqrt{N}$ and $\sqrt{N\sqrt{T}}$, respectively. We also develop two corresponding bootstrap procedures for practical inference, and formally establish their validity. The suggested methods are of practical interest since they require weaker sample size growth conditions than standard fixed-effect quantile regression, and accommodate large $N$ settings. Numerical simulations and an application to mutual fund performance illustrate the proposed methods and the heterogeneity patterns they reveal across quantiles.

Problem

• quantile
• heterogeneous coefficients
• panel data
• individual effects
• asymptotic inference

Innovation

• individual heterogeneous coefficients
• panel quantile regression
• two-step estimation
• bootstrap inference
• asymptotic theory