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(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 .
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(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 .
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This paper studies
We study
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and bias of just-identified instrumental variable quantile regression (IVQR) estimators , nesting order statistics and classical
of conditional quantile estimators such as classical and IV
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We propose a theoretical framework for analyzing small sample properties
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
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with a H\"older continuous process . Using this approximation , we derive remainder bounds with nearly-optimal rates for the
by a H\"older-continuous process with a negligible error. For any consistent estimator, this approximation leads to
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of exact and k-step estimators of IVQR models. Furthermore, we derive a bias formula for exact IVQR
with nearly optimal rates. Second, we study the higher-order bias of exact quantile
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order
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The bias contains
Using a novel non-smooth calculus technique, we uncover previously unknown non-negligible bias
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particular numerical
employed
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novel 1-step adjustment of the estimator
"symmetric" bias correction
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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
implementation. Our simulations confirm the empirical importance of bias correction