Synthetic control methods (SCMs) are a canonical approach used to estimate treatment effects from panel data in the internet economy. 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 -- 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 recommender system 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 an 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：合成控制方法（SCMs）是互联网经济中用于从面板数据估计处理效应的经典方法。我们揭示了一个在SCMs中经常被忽视但普遍存在的“重叠”假设：一个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，并且偏好不同干预措施的单元之间存在足够大的异质性，则重叠假设将不成立。我们通过提出一种推荐系统来解决这一问题，该系统激励具有不同偏好的单元接受它们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，提出了一种SCM，通过向单元提供激励相容的干预推荐，激励在面板数据设置中进行探索。我们证明该估计器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置，其目标是生成所有干预（而不仅仅是控制）下的反事实结果。最后，我们提供了两种假设检验，用于确定给定面板数据集是否满足单元重叠假设。