Asymptotic Properties of Synthetic Control Method

Abstract
This paper provides new insights into the asymptotic properties of the synthetic control method (SCM).We first show that the synthetic control (SC) weight converges to a limiting weight that minimizes the mean squared prediction risk of the treatment-effect estimator when the number of pretreatment periods goes to infinity, and we also quantify the rate of convergence. This result confirms the finding in Ferman and Pinto (2021) that the SC estimator is generally biased under imperfect pretreatment fit, but it also suggests the presence of a bias-variance trade-off in this situation. Next, observing the link between the SCM and model averaging, we establish the asymptotic optimality of the SC estimator under imperfect pretreatment fit, in the sense that it achieves the lowest possible (expected) squared prediction error among all possible treatment effect estimators that are based on a (weighted) average of control units, such as matching, inverse probability weighting and difference-in-differences estimators. The asymptotic optimality holds regardless of whether the number of control units is fixed or divergent. Thus, our results provide justifications for the SCM in a wide range of applications. The theoretical results are verified via simulations