Asymptotically Unbiased Synthetic Control Methods by Moment Matching

TL;DR

This paper tackles endogeneity-induced bias in synthetic control methods by shifting from a strict linearity assumption to a distributional, mixture-model framework. It introduces MMSCM, a moment-matching estimator that identifies SC weights by aligning moments of the treated unit's outcome with a weighted mixture of untreated units' distributions, achieving asymptotic unbiasedness under the mixture assumption. The authors establish convergence and provide inference via conformal methods, plus a bootstrap-like approach to estimate distributional treatment effects. Empirical and simulation results show MMSCM outperforms traditional SCMs and connects to distributional SCMs and optimal transport literature, broadening the applicability of SCMs to policy evaluation with richer counterfactual distributions.

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 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 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 distributions 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.

Figures (9)

• Figure 1: Illustration of SCMs
• Figure 2: Results of simulation studies. Each result represents estimation error ($\tau_{0, t} - \widehat{\tau}$ for an estimator $\widehat{\tau}$). The orange horizontal line denotes the median, and the triangle mark denotes the mean.
• Figure 3: Results of simulation studies on ATE prediction. The $x$-axis is the number of untreated units $J$, and the $y$-axis is the mean squared prediction error.
• Figure 4: Results of simulation studies on distributional treatment effect prediction. The $x$-axis is the number of untreated units $J$, and the $y$-axis is the MMD.
• Figure 5: Counterfactual outcomes in empirical analysis.

Theorems & Definitions (17)

• Definition 2.1
• Proposition 4.4: Proposition 1 in Ferman2021
• Theorem 4.5
• Remark 5.1: Distribution SCMs using divergence or distance minimization.
• Remark 5.2: From CHF matching to moment matching
• Remark 5.3: Reports in Gunsilius2020
• Remark 5.6: Multiple treatments
• Theorem 5.8
• Corollary 5.9
• Remark 5.10: Section 4.1 in Gunsilius2020