  Understanding and characterizing treatment effect variation in randomized
experiments has become essential for going beyond the "black box" of the
average treatment effect. Nonetheless, traditional statistical approaches often
ignore or assume away such variation. In the context of randomized experiments,
this paper proposes a framework for decomposing overall treatment effect
variation into a systematic component explained by observed covariates and a
remaining idiosyncratic component. Our framework is fully randomization-based,
with estimates of treatment effect variation that are entirely justified by the
randomization itself. Our framework can also account for noncompliance, which
is an important practical complication. We make several contributions. First,
we show that randomization-based estimates of systematic variation are very
similar in form to estimates from fully-interacted linear regression and two
stage least squares. Second, we use these estimators to develop an omnibus test
for systematic treatment effect variation, both with and without noncompliance.
Third, we propose an $R^2$-like measure of treatment effect variation explained
by covariates and, when applicable, noncompliance. Finally, we assess these
methods via simulation studies and apply them to the Head Start Impact Study, a
large-scale randomized experiment.
