  We consider the setting of synthetic control methods (SCMs), a canonical
approach used to estimate the treatment effect on the treated in a panel data
setting. We shed light on a frequently overlooked but ubiquitous assumption
made in SCMs of "overlap": a treated unit can be written as some combination --
typically, convex or linear combination -- of the units that remain under
control. We show that if units select their own interventions, and there is
sufficiently large heterogeneity between units that prefer different
interventions, overlap will not hold. We address this issue by proposing a
framework which incentivizes units with different preferences to take
interventions they would not normally consider. Specifically, leveraging tools
from information design and online learning, we propose a SCM that incentivizes
exploration in panel data settings by providing incentive-compatible
intervention recommendations to units. We establish this estimator obtains
valid counterfactual estimates without the need for an a priori overlap
assumption. We extend our results to the setting of synthetic interventions,
where the goal is to produce counterfactual outcomes under all interventions,
not just control. Finally, we provide two hypothesis tests for determining
whether unit overlap holds for a given panel dataset.
