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

This paper proposes estimation and inference procedures for quantiles of the heterogeneous individual-specific coefficients in panel data. 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 the 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 empirical application illustrate the empirical effectiveness of the methods under both designs.