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

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 $\tau$-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.