Theory for identification and Inference with Synthetic Controls: A Proximal Causal Inference Framework

Synthetic control methods are commonly used to estimate the treatment effect on a single treated unit in panel data settings. A synthetic control (SC) is 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. A common practice to estimate the SC weights is to regress the pre-treatment outcome process of the treated unit on that of control units. However, it has been established that such regression estimators can fail to be consistent. In addition, formal statistical inference is challenging under the SC framework. In this paper, building upon Miao, Geng, and Tchetgen Tchetgen (2018) and Tchetgen Tchetgen et al. (2020), we introduce a proximal causal inference framework for the SC approach and formalize identification and inference for both the SC weights and the treatment effect on the treated unit. We show that the outcomes of control units previously perceived as unusable can be repurposed to identify and consistently estimate the SC weights. We also propose to view the difference in the post-treatment outcomes between the treated unit and the SC as a time series with the treatment effect captured by a deterministic time trend, which opens the door to a rich and extensive literature on time-series analysis for estimation of the treatment effect. We further extend the traditional linear interactive fixed effects model to accommodate general nonlinear models allowing for binary and count outcomes which are currently understudied in the SC literature. We illustrate our proposed methods with simulation studies and an application to the evaluation of the 1990 German Reunification.