Working Paper

Conditional Quantile Estimators: A Small Sample Theory

Grigory Franguridi, Bulat Gafarov, Kaspar Wüthrich
CESifo, Munich, 2021

CESifo Working Paper No. 9046

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 Hölder-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 (1/n).
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.

CESifo Category
Empirical and Theoretical Methods
Keywords: non-smooth estimators, KMT coupling, Hungarian construction, higher-order asymptotic distribution, higher-order stochastic expansion, order statistic, bias correction, mixed integer linear programming (MILP), exact estimators, k-step estimators, quantile
JEL Classification: C210, C260