Synthetic Control Methods (SCMs) have become an essential tool for comparative case studies. The fundamental idea of SCMs is to estimate the counterfactual outcomes of a treated unit using a weighted sum of the observed outcomes of untreated units. The accuracy of the synthetic control (SC) is critical for evaluating the treatment effect of a policy intervention; therefore, the estimation of SC weights has been the focus of extensive research. In this study, we first point out that existing SCMs suffer from an endogeneity problem, the correlation between the outcomes of untreated units and the error term of the synthetic control, which yields a bias in the treatment effect estimator. We then propose a novel SCM based on density matching, assuming that the density of outcomes of the treated unit can be approximated by a weighted average of the joint density of untreated units (i.e., a mixture model). Based on this assumption, we estimate SC weights by matching the moments of treated outcomes with the weighted sum of moments of untreated outcomes. Our proposed method has three advantages over existing methods: first, our estimator is asymptotically unbiased under the assumption of the mixture model; second, due to the asymptotic unbiasedness, we can reduce the mean squared error in counterfactual predictions; third, our method generates full densities of the treatment effect, not merely expected values, which broadens the applicability of SCMs. We provide experimental results to demonstrate the effectiveness of our proposed method.

翻译：合成控制方法(SCMs)已成为比较案例研究的重要工具。SCMs的基本思想是利用未处理单元观测结果的加权和来估计处理单元的反事实结果。合成控制(SC)的准确性对于评估政策干预的处理效应至关重要，因此SC权重的估计一直是广泛研究的焦点。本研究首先指出现有SCMs存在内生性问题——未处理单元结果与合成控制误差项之间的相关性——这会导致处理效应估计量产生偏差。接着我们提出一种基于密度匹配的新型SCMs，其假设处理单元结果的密度可通过未处理单元联合密度的加权平均（即混合模型）来近似。基于该假设，我们通过匹配处理结果矩与未处理结果矩的加权和来估计SC权重。与现有方法相比，所提方法具有三个优势：首先，在混合模型假设下我们的估计量是渐近无偏的；其次，由于渐近无偏性，我们可降低反事实预测的均方误差；第三，我们的方法能生成处理效应的完整密度分布而不仅是期望值，从而拓展了SCMs的适用性。我们通过实验结果证明了所提方法的有效性。