Minimum Distance Estimation of Quantile Panel Data Models

We develop a two-stage minimum distance estimator for quantile panel data models with possibly correlated unit effects. In the first stage, we estimate group-specific quantile regressions to obtain fitted conditional quantiles; in the second stage, we use GMM to project these fitted values on individual and group regressors. Because the convergence rate depends on the presence of unobserved group heterogeneity, we provide adaptive estimation and inference that are uniformly valid in the degree of heterogeneity. An analytical variance estimator consistently captures the leading component without requiring pretesting or selection. Under the same identification conditions as existing grouped instrumental variable quantile methods, we demonstrate that the procedure yields higher precision, which is confirmed by simulations and empirical evidence. In an application to the Food Stamp Program, we find significant birth weight gains at the lower tail of the distribution, illustrating the method’s ability to capture heterogeneous effects.