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, with weights typically estimated by regressing pre-treatment outcomes of the treated unit to those of the control units. However, it has been established that such regression estimators can fail to be consistent. In this paper, we introduce a proximal causal inference framework to formalize identification and inference for both the SC weights and the treatment effect on the treated. We show that control units previously perceived as unusable can be repurposed to 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, which opens the door to a rich literature on time-series analysis for treatment effect estimation. We further extend the traditional linear model to accommodate general nonlinear models allowing for binary and count outcomes which are understudied in the SC literature. We illustrate our proposed methods with simulation studies and an application to evaluation of the 1990 German Reunification.