  In this paper, we propose a class of low-rank panel quantile regression
models which allow for unobserved slope heterogeneity over both individuals and
time. We estimate the heterogeneous intercept and slope matrices via nuclear
norm regularization followed by sample splitting, row- and column-wise quantile
regressions and debiasing. We show that the estimators of the factors and
factor loadings associated with the intercept and slope matrices are
asymptotically normally distributed. In addition, we develop two specification
tests: one for the null hypothesis that the slope coefficient is a constant
over time and/or individuals under the case that true rank of slope matrix
equals one, and the other for the null hypothesis that the slope coefficient
exhibits an additive structure under the case that the true rank of slope
matrix equals two. We illustrate the finite sample performance of estimation
and inference via Monte Carlo simulations and real datasets.
