  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 (summaries of) pre-treatment outcomes and
measured covariates of the treated unit to those of the control units. However,
it has been established that in the absence of a good fit, such regression
estimator will generally perform poorly. 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. We view the difference in
the post-treatment outcomes between the treated unit and the SC as a time
series, which opens the door to various time series methods for treatment
effect estimation. The proposed framework can accommodate nonlinear models,
which allows for binary and count outcomes that are understudied in the SC
literature. We illustrate with simulation studies and an application to
evaluation of the 1990 German Reunification.
