Asymptotically Unbiased Synthetic Control Methods by Density Matching

2023-07-20

AI Summary

Existing synthetic control methods (SCM) suffer from endogenous bias in counterfactual prediction and treatment effect estimation due to unobserved confounding. This paper proposes a novel weighting framework based on density matching, introducing— for the first time in SCM—the density mixture assumption. By matching the joint distribution moments of treated and control units, the method achieves asymptotically unbiased counterfactual prediction. The approach combines theoretical rigor with computational feasibility: we formally establish the asymptotic unbiasedness of the estimator and its mean squared error (MSE) improvement over conventional SCM; it transcends traditional point estimation by delivering the full density function of the treatment effect; and empirical results demonstrate substantial gains in counterfactual prediction accuracy while effectively characterizing the distributional features of treatment effects.

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.

Problem

• Bias in Synthetic Control Methods
• Endogeneity issue in SCMs
• Matching densities for unbiased estimation

Innovation

• Density matching for synthetic control
• Asymptotically unbiased estimator
• Full treatment effect densities