This paper examines regression-adjusted estimation and inference of unconditional quantile treatment eﬀects (QTEs) under covariate-adaptive randomizations (CARs). Datasets from ﬁeld experiments usually contain extra baseline covariates in addition to the strata indicators. We propose to incorporate these extra covariates via auxiliary regressions in the estimation and inference of unconditional QTEs. We establish the consistency, limit distribution, and validity of the multiplier bootstrap of the QTE estimator under CARs. The auxiliary regression may be estimated parametrically, nonparametrically, or via regularization when the data are high-dimensional. Even when the auxiliary regression is misspeciﬁed, the proposed bootstrap inferential procedure still achieves the nominal rejection probability in the limit under the null. When the auxiliary regression is correctly speciﬁed, the regression-adjusted estimator achieves the minimum asymptotic variance. We also derive the optimal pseudo true values for the potentially misspeciﬁed parametric model that minimize the asymptotic variance of the corresponding QTE estimator. Our estimation and inferential methods can be implemented without tuning parameters and they allow for common choices of auxiliary regressions such as linear, probit and logit regressions despite the fact that these regressions may be misspeciﬁed. Finite-sample performance of the new estimation and inferential methods is assessed in simulations and an empirical application studying the impact of child health and nutrition on educational outcomes is included.
This paper examines methods of inference concerning quantile treatment eﬀects (QTEs) in randomized experiments with matched-pairs designs (MPDs). We derive the limit distribution of the QTE estimator under MPDs, highlighting the diﬀiculties that arise in analytical inference due to parameter tuning. We show that the naïve weighted bootstrap fails to approximate the limit distribution of the QTE estimator under MPDs because it ignores the dependence structure within the matched pairs.To address this diﬀiculty we propose two bootstrap methods that can consistently approximate the limit distribution: the gradient bootstrap and the weighted bootstrap of the inverse propensity score weighted (IPW) estimator. The gradient bootstrap is free of tuning parameters but requires knowledge of the pair identities. The weighted bootstrap of the IPW estimator does not require such knowledge but involves one tuning parameter. Both methods are straightforward to implement and able to provide pointwise conﬁdence intervals and uniform conﬁdence bands that achieve exact limiting coverage rates. We demonstrate their ﬁnite sample performance using simulations and provide an empirical application to a well-known dataset in microﬁnance.