Synthetic control methods (SCMs) have become a crucial tool for causal inference in comparative case studies. The fundamental idea of SCMs is to estimate counterfactual outcomes for a treated unit by using a weighted sum of observed outcomes from untreated units. The accuracy of the synthetic control (SC) is critical for estimating the causal effect, and hence, the estimation of SC weights has been the focus of much research. In this paper, we first point out that existing SCMs suffer from an implicit endogeneity problem, which is the correlation between the outcomes of untreated units and the error term in the model of a counterfactual outcome. We show that this problem yields a bias in the causal 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 densities of untreated units (i.e., a mixture model). Based on this assumption, we estimate SC weights by matching moments of treated outcomes and 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 for counterfactual prediction. Third, our method generates full densities of the treatment effect, not only expected values, which broadens the applicability of SCMs. We provide experimental results to demonstrate the effectiveness of our proposed method.

翻译：合成控制方法已成为比较案例研究中因果推断的关键工具。其核心思想是通过对未处理单位的观测结果进行加权求和，来估计处理单位的反事实结果。合成控制的准确性对因果效应的估计至关重要，因此合成控制权重的估计一直是研究的重点。本文首先指出现有合成控制方法存在隐式内生性问题，即未处理单位的结果与反事实结果模型中的误差项存在相关性。我们证明该问题会导致因果效应估计产生偏差。随后，我们提出一种基于密度匹配的新型合成控制方法，假设处理单位的结果密度可通过未处理单位结果密度的加权平均（即混合模型）近似。基于该假设，我们通过匹配处理单位结果的矩与未处理单位结果矩的加权和来估计合成控制权重。相比现有方法，本方法具有三大优势：第一，在混合模型假设下，估计量具有渐近无偏性；第二，利用渐近无偏性可降低反事实预测的均方误差；第三，该方法不仅能生成期望值，还能生成处理效应的完整密度分布，从而拓展了合成控制方法的适用性。我们通过实验验证了所提方法的有效性。