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，综合控制（SC）方法已迅速成为面板数据观测研究中评估因果效应的主流方法之一。现有理论讨论通常基于潜在结果由因子模型生成的假设对SC方法进行公式化论证。本文从设计视角出发，假设处理单元与处理时点的选择机制存在模型化表述，对SC方法展开研究。我们证明，在随机分配条件下，标准SC估计量通常存在偏误。为此提出修正无偏综合控制（MUSC）估计量，该估计量在随机分配条件下保证无偏性，并推导出其精确的随机化有限样本方差，同时给出该方差的无偏估计量。基于真实数据的实证表明，在随机分配条件下，SC类估计量的均方根误差显著低于其他常用估计量。我们证明，当处理时点与其他时期具有相似性时（例如处理时点随机选取），此类改进效果具有弱保证性。尽管研究结论直接适用于处理随机分配的设定，但我们认为其对于观测研究中的模型驱动方法仍具有补充价值。