Primary Submission Category: Difference in Differences, Synthetic Control, Methods for Panel and Longitudinal Data
Inference for synthetic controls via refined placebo tests
Authors: Lihua Lei, Timothy Sudijono,
Presenting Author: Lihua Lei*
The synthetic control method is often applied to problems with one treated unit and a small number of control units. A common inferential task in this setting is to test null hypotheses regarding the average treatment effect on the treated. 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 implemented in practice. An alternative is permutation inference, which is related to a common diagnostic called the placebo test. It has provable Type-I error guarantees in finite samples without simplification of the method, when the treatment is uniformly assigned. However, it often has low power at a common level like α=0.05 when N is small. We propose a novel leave-two-out procedure that bypasses this issue, while still maintaining the same finite-sample Type-I error guarantee under uniform assignment for a wide range of N. Unlike the placebo test whose Type-I error always equals the theoretical upper bound, our procedure often achieves a lower unconditional Type-I error than theory suggests; this enables useful inference in the challenging regime when α<1/N. Empirically, our procedure achieves a higher power when the effect size is reasonably large and a comparable power otherwise. We generalize our procedure to non-uniform assignments and show how to conduct sensitivity analysis.