Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control (SC) methods have quickly become one of the leading methods for estimating causal effects in observational studies in settings with panel data. Formal discussions often motivate SC methods by the assumption that the potential outcomes were generated by a factor model. Here we study SC methods from a design-based perspective, assuming a model for the selection of the treated unit(s) and period(s). We show that the standard SC estimator is generally biased under random assignment. We propose a Modified Unbiased Synthetic Control (MUSC) estimator that guarantees unbiasedness under random assignment and derive its exact, randomization-based, finite-sample variance. We also propose an unbiased estimator for this variance. We document in settings with real data that under random assignment, SC-type estimators can have root mean-squared errors that are substantially lower than that of other common estimators. We show that such an improvement is weakly guaranteed if the treated period is similar to the other periods, for example, if the treated period was randomly selected. While our results only directly apply in settings where treatment is assigned randomly, we believe that they can complement model-based approaches even for observational studies.

翻译：自Abadie和Gardeazabal（2003）提出以来，合成控制（SC）方法迅速成为面板数据情境下观测研究中评估因果效应的主要方法之一。形式化讨论通常基于潜在结果由因子模型生成的假设来推导SC方法。本文从基于设计的视角研究SC方法，即假定处理单元和时期的选择遵循某个模型。我们证明，在随机分配条件下，标准SC估计量通常存在偏误。我们提出了一种修正无偏合成控制（MUSC）估计量，该估计量可保证在随机分配下的无偏性，并推导出其精确的、基于随机化的有限样本方差。同时，我们还提出了该方差的无偏估计量。基于真实数据的实证表明，在随机分配场景下，SC类估计量的均方根误差可能显著低于其他常见估计量。我们证明，当处理期与其他时期相似（例如处理期被随机选择）时，这种改进具有弱保障性。尽管我们的结论仅直接适用于处理变量随机分配的设定，但我们认为，即使在观测研究中，这些结论也可补充基于模型的方法。