We aim to evaluate the causal impact of an intervention when time series data on a single treated unit and multiple untreated units are available, in pre- and post- treatment periods. In their seminal work, Abadie and Gardeazabal (2003) and Abadie et al. (2010) proposed a synthetic control (SC) method as an approach to relax the parallel trend assumption on which difference-in-differences methods typically rely upon. The term "synthetic control" refers to a weighted average of control units built to match the treated unit's pre-treatment outcome trajectory, such that the SC's post-treatment outcome predicts the treated unit's unobserved potential outcome under no treatment. The treatment effect is then estimated as the difference in post-treatment outcomes between the treated unit and the SC. A common practice to estimate the weights is to regress the pre-treatment outcome process of the treated unit on that of control units using ordinary or weighted (constrained) least squares. However, it has been established that these estimators can fail to be consistent under standard time series asymptotic regimes. Furthermore, inferences with synthetic controls are typically performed by placebo test which lacks formal justification. In this paper, we introduce a proximal causal inference framework for the synthetic control approach and formalize identification and inference for both the synthetic control weights and the treatment effect on the treated unit. We further extend the traditional linear interactive fixed effects model to accommodate more general nonlinear models allowing for binary and count outcomes which are currently under-studied in the synthetic control literature. We illustrate our proposed methods with simulation studies and an application to the evaluation of the 1990 German Reunification.
翻译:在治疗前和治疗后阶段,当获得关于单一处理过的单位和多个未经处理的单位的时间序列数据时,我们的目标是评估干预的因果关系。在其开创性工作中,Abadie和Gardeazabal(2003年)和Abadie等人(2010年)提出了合成控制方法(SC),以此作为一种方法,以放松使用差异差异方法通常依赖的平行趋势假设。“合成控制”一词是指为与处理过的单位的预处理结果轨迹相匹配而建造的控制单位的加权平均值,因此SC的后处理结果预测了处理过的单位在不治疗下无法观察到的潜在结果。随后,治疗效果作为处理后的后结果差异,Abadie和Gardeazazabal(2003年)和Abadie等人(2010年)提出了合成控制方法。一种通常的做法,即用普通的或加权的(受限制的)最低方位表示控制单位的预处理结果。但是,这些估算器目前可能无法在标准的时间序列中保持一致,因为理事会的后期处理结果是未经任何处理的。此外,我们通常采用合成控制模式进行定期分析的结果,因此,在进行这种结果测试时,因此,我们先进行这种分析时,在正式分析后进行。