Statistical Inference for Average Treatment Effects Estimated by Synthetic Control Methods

The synthetic control (SC) method, a powerful tool for estimating average treatment effects (ATE), is increasingly popular in fields such as statistics, economics, political science, and marketing. The SC is particularly suitable for estimating ATE with a single (or a few) treated unit(s), a fixed number of control units, and large pre and post-treatment periods (which we refer as "long panels"). To date, there has been no formal inference theory for SC ATE estimator with long panels under general conditions. Existing work mostly use placebo tests for inference or some permutation methods when the post-treatment period is small. In this article, we derive the asymptotic distribution of the SC and modified synthetic control (MSC) ATE estimators using projection theory. We show that a properly designed subsampling method can be used to obtain confidence intervals and conduct inference whereas the standard bootstrap cannot. Simulations and an empirical application that examines the effect of opening a physical showroom by an e-tailer demonstrate the usefulness of the MSC method in applications. Supplementary materials for this article are available online.