Primary Submission Category: Synthetic Control Method
Inference for Synthetic Controls via Refined Placebo Tests
Authors: Timothy Sudijono, Lihua Lei,
Presenting Author: Timothy Sudijono*
The synthetic control method is often applied to problems with one treated unit and a small number of control units. Inference procedures that are justified asymptotically are often unsatisfactory due to (1) small sample sizes that render large-sample approximation fragile and (2) simplification of the estimation procedure that is actually implemented in practice. An alternative is design-based inference, which is closely related to the placebo test, a widely used diagnostic tool in practice. It provides valid Type-I error control in finite samples without artificial simplifications of the method when the treatment is assigned uniformly among units. Despite this robustness, it suffers from low resolution since the null distribution is constructed from only $N$ reference estimates, where $N$ is the sample size. Inspired by a connection to the conformal inference literature, we propose a novel leave-two-out procedure that bypasses this issue, providing $O(N^2)$ reference estimates while still maintaining finite-sample Type-I error control under uniform assignments. Unlike the placebo test whose Type-I error always equals the theoretical upper bound, our procedure often achieves a lower Type-I error than theory suggests and a higher power when the effect size is reasonably large. To account for deviation from uniform assignments, we generalize our procedure to allow for non-uniform assignments and show how to conduct sensitivity analysis based on quadratic programming.