A Method of Moments Approach to Asymptotically Unbiased Synthetic Controls

TL;DR

This paper introduces a General Method of Moments approach for constructing Synthetic Controls that achieves asymptotic unbiasedness with fixed control units by using other units as instruments, improving upon existing methods.

Contribution

The paper proposes a novel GMM-based Synthetic Control method that remains asymptotically unbiased with fixed controls, unlike traditional approaches.

Findings

The new method is asymptotically unbiased as pre-treatment periods increase.

It allows consistent estimation of average treatment effects with increasing time periods.

Simulation and empirical results show improved performance over existing methods.

Abstract

A common approach to constructing a Synthetic Control unit is to fit on the outcome variable and covariates in pre-treatment time periods, but it has been shown by Ferman and Pinto (2019) that this approach does not provide asymptotic unbiasedness when the fit is imperfect and the number of controls is fixed. Many related panel methods have a similar limitation when the number of units is fixed. I introduce and evaluate a new method in which the Synthetic Control is constructed using a General Method of Moments approach where units not being included in the Synthetic Control are used as instruments. I show that a Synthetic Control Estimator of this form will be asymptotically unbiased as the number of pre-treatment time periods goes to infinity, even when pre-treatment fit is imperfect and the number of units is fixed. Furthermore, if both the number of pre-treatment and post-treatment time periods go to infinity, then averages of treatment effects can be consistently estimated. I conduct simulations and an empirical application to compare the performance of this method with existing approaches in the literature.