Conditional quantile estimators: A small sample theory

This paper studies the small sample properties and bias of just-identified instrumental variable quantile regression (IVQR) estimators, nesting order statistics and classical quantile regression. We propose a theoretical framework for analyzing small sample properties based on a novel approximation of the discontinuous sample moments with a H\"older continuous process. Using this approximation, we derive remainder bounds with nearly-optimal rates for the asymptotic linear expansions of exact and k-step estimators of IVQR models. Furthermore, we derive a bias formula for exact IVQR estimators up to order $O\left(\frac{1}{n}\right)$. The bias contains components that cannot be consistently estimated and depend on the particular numerical estimation algorithm. To circumvent this problem, we propose a novel 1-step adjustment of the estimator, which admits a feasible bias correction. We suggest using bias-corrected exact estimators, when possible, to achieve the smallest bias. Otherwise, applying 1-step corrections may improve the higher-order bias and MSE of any consistent estimator.