Asymptotically unbiased synthetic control methods by moment matching

Abstract

Synthetic control methods have become a fundamental tool for comparative case studies. The core idea behind synthetic control methods is to estimate treatment effects by predicting counterfactual outcomes for a treated unit using a weighted combination of observed outcomes from untreated units. The accuracy of these predictions is crucial for evaluating the treatment effect of a policy intervention. Subsequent research has therefore focused on estimating synthetic control weights. In this study, we highlight a key endogeneity issue in existing synthetic control methods; that is, the correlation between the outcomes of untreated units and the error term of the synthetic control, which leads to bias in both counterfactual outcome prediction and treatment effect estimation. To address this issue, we propose a novel synthetic control method based on moment matching, assuming that the outcome distribution of the treated unit can be approximated by a weighted mixture of the distributions of untreated units. Under this assumption, we estimate synthetic control weights by matching the moments of the treated outcomes with the weighted sum of the moments of the untreated outcomes. Our method offers three advantages: first, under the mixture model assumption, our estimator is asymptotically unbiased; second, this asymptotic unbiasedness reduces the mean squared error in counterfactual predictions; and third, our method provides full distributions of the treatment effect rather than just expected values, thereby broadening the applicability of synthetic control methods. Finally, we present experimental results that demonstrate the effectiveness of our approach.