Panel Quantile-on-Quantile Regression

Standard panel quantile regression estimates how the θ-th quantile of Y depends on X, treating X as a fixed covariate. The QQ extension makes the slope itself a function of τ, the quantile rank of X, revealing how the dependence changes as X moves through its own distribution — something a single quantile regression cannot show.

A common starting point is a Gaussian kernel with bandwidth h = 0.1 (on the [0,1] quantile scale). You should report results for at least two additional bandwidth values (e.g., 0.05 and 0.15) to show that conclusions do not hinge on this choice. Cross-validation criteria adapted for quantile loss have also been proposed.

As a practical guide, aim for N ≥ 15 and T ≥ 15 for reasonably stable estimates at the 0.1 and 0.9 quantiles. With smaller panels, restrict the quantile grid to the interior (0.2–0.8) and widen the kernel bandwidth to borrow more data.

The standard QQ framework is bivariate: one outcome quantile regressed on one predictor quantile. For multivariate settings, researchers typically run pairwise QQ regressions or use partial quantile approaches, partialling out controls first before applying QQ to the residuals.

No single canonical package exists. In R, the quantreg and np packages provide building blocks; researchers assemble the estimator manually. Custom MATLAB routines circulate in replication files accompanying published papers. Stata users typically call rqprocess or use user-written ado files combined with panel demeaning.