  This paper studies inference in a randomized controlled trial (RCT) with
covariate-adaptive randomization (CAR) and imperfect compliance of a binary
treatment. In this context, we study inference on the LATE. As in Bugni et al.
(2018,2019), CAR refers to randomization schemes that first stratify according
to baseline covariates and then assign treatment status so as to achieve
``balance'' within each stratum. In contrast to these papers, however, we allow
participants of the RCT to endogenously decide to comply or not with the
assigned treatment status.
  We study the properties of an estimator of the LATE derived from a ``fully
saturated'' IV linear regression, i.e., a linear regression of the outcome on
all indicators for all strata and their interaction with the treatment
decision, with the latter instrumented with the treatment assignment. We show
that the proposed LATE estimator is asymptotically normal, and we characterize
its asymptotic variance in terms of primitives of the problem. We provide
consistent estimators of the standard errors and asymptotically exact
hypothesis tests. In the special case when the target proportion of units
assigned to each treatment does not vary across strata, we can also consider
two other estimators of the LATE, including the one based on the ``strata fixed
effects'' IV linear regression, i.e., a linear regression of the outcome on
indicators for all strata and the treatment decision, with the latter
instrumented with the treatment assignment.
  Our characterization of the asymptotic variance of the LATE estimators allows
us to understand the influence of the parameters of the RCT. We use this to
propose strategies to minimize their asymptotic variance in a hypothetical RCT
based on data from a pilot study. We illustrate the practical relevance of
these results using a simulation study and an empirical application based on
Dupas et al. (2018).
