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.

翻译：我们考虑合成控制方法（SCMs）的设定，这是一种在面板数据环境中估计处理组处理效应的经典方法。我们揭示了SCMs中一个常被忽视但普遍存在的“重叠”假设：被处理单元可以表示为保持控制状态的单元的某种组合——通常是凸组合或线性组合。我们证明，如果单元自行选择干预措施，且偏好不同干预措施的单元之间存在足够大的异质性，那么重叠假设将不成立。为解决此问题，我们提出一个框架，通过激励偏好不同的单元采取其通常不会考虑的干预措施。具体而言，我们利用信息设计和在线学习的工具，提出一种SCM，通过向单元提供激励相容的干预建议，在面板数据环境中激励探索。我们证明该估计量无需先验重叠假设即可获得有效的反事实估计。我们将结果扩展到合成干预的设定，其目标是生成所有干预措施下的反事实结果，而不仅仅是控制状态。最后，我们提供两种假设检验方法，用于判断给定面板数据集是否满足单元重叠假设。