  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.
