The synthetic control method has become a widely popular tool to estimate causal effects with observational data. Despite this, inference for synthetic control methods remains challenging. Often, inferential results rely on linear factor model data generating processes. In this paper, we characterize the conditions on the factor model primitives (the factor loadings) for which the statistical risk minimizers are synthetic controls (in the simplex). Then, we propose a Bayesian alternative to the synthetic control method that preserves the main features of the standard method and provides a new way of doing valid inference. We explore a Bernstein-von Mises style result to link our Bayesian inference to the frequentist inference. For linear factor model frameworks we show that a maximum likelihood estimator (MLE) of the synthetic control weights can consistently estimate the predictive function of the potential outcomes for the treated unit and that our Bayes estimator is asymptotically close to the MLE in the total variation sense. Through simulations, we show that there is convergence between the Bayes and frequentist approach even in sparse settings. Finally, we apply the method to re-visit the study of the economic costs of the German re-unification and the Catalan secession movement. The Bayesian synthetic control method is available in the bsynth R-package.

翻译：合成控制法已成为基于观测数据估计因果效应的广泛流行工具。尽管如此，合成控制法的推断仍具有挑战性。现有推断结果通常依赖于线性因子模型的数据生成过程。本文首先刻画了因子模型基础要素（因子载荷）需满足的条件，在此条件下统计风险最小化器即为单纯形内的合成控制。随后，我们提出一种保留标准方法核心特征并提供有效推断新途径的贝叶斯替代方案。通过探究伯恩斯坦-冯·米塞斯型结果，我们将贝叶斯推断与频率学派推断联系起来。在线性因子模型框架下，我们证明合成控制权重的极大似然估计可一致估计处理单元潜在结果的预测函数，且贝叶斯估计量在总变差意义下渐近逼近极大似然估计。模拟结果表明，即便在稀疏设定下，贝叶斯方法与频率学派方法仍具有收敛性。最后，我们将该方法应用于重新审视德国统一与加泰罗尼亚分离运动的经济成本研究。贝叶斯合成控制法可通过bsynth R语言包实现。