  We study the bias of classical quantile regression and instrumental variable
quantile regression estimators. While being asymptotically first-order
unbiased, these estimators can have non-negligible second-order biases. We
derive a higher-order stochastic expansion of these estimators using empirical
process theory. Based on this expansion, we derive an explicit formula for the
second-order bias and propose a feasible bias correction procedure that uses
finite-difference estimators of the bias components. The proposed bias
correction method performs well in simulations. We provide an empirical
illustration using Engel's classical data on household expenditure.
