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.

翻译：我们研究了经典分位数回归和工具变量分位数回归估计量的偏差。尽管这些估计量在渐近一阶无偏，但它们可能存在不可忽略的二阶偏差。我们利用经验过程理论推导了这些估计量的高阶随机展开。基于该展开，我们得到了二阶偏差的显式公式，并提出了一种可行的偏差修正方法，该方法使用有限差分估计量来估计偏差分量。所提出的偏差修正方法在模拟中表现良好。我们利用恩格尔关于家庭支出的经典数据进行了实证说明。