  Datasets from field experiments with covariate-adaptive randomizations (CARs)
usually contain extra covariates in addition to the strata indicators. We
propose to incorporate these additional covariates via auxiliary regressions in
the estimation and inference of unconditional quantile treatment effects (QTEs)
under CARs. We establish the consistency and limit distribution of the
regression-adjusted QTE estimator and prove that the use of multiplier
bootstrap inference is non-conservative 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 discuss forms of adjustments that can
improve the efficiency of the QTE estimators. The finite sample performance of
the new estimation and inferential methods is studied in simulations and an
empirical application to a well-known dataset concerned with expanding access
to basic bank accounts on savings is reported.
