Asymptotic Properties of the Synthetic Control Method

TL;DR

This paper investigates the asymptotic behavior of the synthetic control method, showing its convergence properties and optimality in prediction error, thus providing theoretical justification for its use in various applications.

Contribution

It establishes the limiting behavior and optimality of the synthetic control estimator, linking it to model averaging and comparing it to other methods.

Findings

SC weight converges to a risk-minimizing limit

SC estimator is asymptotically optimal under imperfect fit

Results are validated through simulations

Abstract

This paper provides new insights into the asymptotic properties of the synthetic control method (SCM). We 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. Observing the link between the SCM and model averaging, we further establish the asymptotic optimality of the SC estimator under imperfect pretreatment fit, in the sense that it achieves the lowest possible squared prediction error among all possible treatment effect estimators that are based on an average of control units, such as matching, inverse probability weighting and difference-in-differences. 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.