# Asymptotically Unbiased Synthetic Control Methods by Moment Matching

arXiv: https://arxiv.org/abs/2307.11127 · HTML: https://mlsanity.com/paper/2307.11127/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 20 Jul 2023.

## Builds on (most cited first, 14 of 14)

- [Non-Negative Bregman Divergence Minimization for Deep Direct Density Ratio Estimation](https://mlsanity.com/paper/2006.06979/) (2020)
- [Conditional Distributional Treatment Effect with Kernel Conditional Mean Embeddings and U-Statistic Regression](https://mlsanity.com/paper/2102.08208/) (2021)
- [An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls](https://mlsanity.com/paper/1712.09089/) (2017)
- [The Augmented Synthetic Control Method](https://mlsanity.com/paper/1811.04170/) (2018)
- [Robust and Agnostic Learning of Conditional Distributional Treatment Effects](https://mlsanity.com/paper/2205.11486/) (2022)
- [On the Assumptions of Synthetic Control Methods](https://mlsanity.com/paper/2112.05671/) (2021)
- [Distributional synthetic controls](https://mlsanity.com/paper/2001.06118/) (2020)
- [Feature Selection for Discovering Distributional Treatment Effect Modifiers](https://mlsanity.com/paper/2206.00516/) (2022)
- [Randomization Tests in Observational Studies with Staggered Adoption of Treatment](https://mlsanity.com/paper/1912.10610/) (2019)
- [Double and Single Descent in Causal Inference with an Application to High-Dimensional Synthetic Control](https://mlsanity.com/paper/2305.00700/) (2023)
- [Prediction Intervals for Synthetic Control Methods](https://mlsanity.com/paper/1912.07120/) (2019)
- [Uncertainty Quantification in Synthetic Controls with Staggered Treatment Adoption](https://mlsanity.com/paper/2210.05026/) (2022)
- [On the Misspecification of Linear Assumptions in Synthetic Control](https://mlsanity.com/paper/2302.12777/) (2023)
- [scpi: Uncertainty Quantification for Synthetic Control Methods](https://mlsanity.com/paper/2202.05984/) (2022)

## Built on by (most cited first, 2 of 2)

- [Constructing Synthetic Treatment Groups without the Mean Exchangeability Assumption](https://mlsanity.com/paper/2309.16409/) (2023)
- [Counterexamples and Sufficient Conditions: Comments on "Optimally-Transported Generalized Method of Moments"](https://mlsanity.com/paper/2609.20260/) (2026)

## Abstract (arXiv)

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.