Conditional quantile estimators: A small sample theory

We study the small sample properties of conditional quantile estimators such as classical and IV quantile regression. First, we propose a higher-order analytical framework for comparing competing estimators in small samples and assessing the accuracy of common inference procedures. Our framework is based on a novel approximation of the discontinuous sample moments by a H\"older-continuous process with a negligible error. For any consistent estimator, this approximation leads to asymptotic linear expansions with nearly optimal rates. Second, we study the higher-order bias of exact quantile estimators up to $O\left(\frac{1}{n}\right)$. Using a novel non-smooth calculus technique, we uncover previously unknown non-negligible bias components that cannot be consistently estimated and depend on the employed estimation algorithm. To circumvent this problem, we propose a "symmetric" bias correction, which admits a feasible implementation. Our simulations confirm the empirical importance of bias correction.