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

Synthetic control (SC) methods are commonly used to estimate the treatment effect on a single treated unit in panel data settings. An SC is a weighted average of control units built to match the treated unit, with weights typically estimated by regressing pre-treatment outcomes and measured covariates of the treated unit to those of the control units. However, the classical SC method was primarily proposed for empirical settings where a good pre-treatment fit is attainable (Abadie et al., 2010). In this paper, we introduce a proximal causal inference framework to formalize identification and inference for both the SC and ultimately the treatment effect on the treated, based on the observation that control units not contributing to the construction of an SC can be repurposed as proxies of latent confounders, thus extending the applicability of SC methods to cases where the pre-treatment fit is poor even with many pre-treatment periods. We show that several existing uncertainty quantification methods of treatment effect for the classical SC methods can be adapted to the proximal inference approach. The proposed framework can accommodate nonlinear models, which allows for binary and count outcomes both of which remain understudied in the SC literature. We illustrate with comprehensive simulation studies and an application to the evaluation of the 1990 German Reunification.