Asymptotically Unbiased Synthetic Control Methods by Density Matching

Abstract: Synthetic Control Methods (SCMs) have become a fundamental tool for comparative case studies. The core idea behind SCMs 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 SC weights. In this study, we highlight a key endogeneity issue in existing SCMs-namely, 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 SCM based on density matching, assuming that the outcome density of the treated unit can be approximated by a weighted mixture of the joint density of untreated units. Under this assumption, we estimate SC 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 densities of the treatment effect rather than just expected values, thereby broadening the applicability of SCMs. Finally, we present experimental results that demonstrate the effectiveness of our approach.