We propose a nonlinear difference-in-differences method to estimate multivariate counterfactual distributions in classical treatment and control study designs with observational data. Our approach sheds a new light on existing approaches like the changes-in-changes and the classical semiparametric difference-in-differences estimator and generalizes them to settings with multivariate heterogeneity in the outcomes. The main benefit of this extension is that it allows for arbitrary dependence and heterogeneity in the joint outcomes. We demonstrate its utility both on synthetic and real data. In particular, we revisit the classical Card \& Krueger dataset, examining the effect of a minimum wage increase on employment in fast food restaurants; a reanalysis with our method reveals that restaurants tend to substitute full-time with part-time labor after a minimum wage increase at a faster pace. A previous version of this work was entitled "An optimal transport approach to causal inference.

翻译：我们提出一种非线性差分中的方法，用于在基于观测数据的经典处理组-对照组研究设计中估计多元反事实分布。本方法为现有方法（如变化-变化法及经典半参数差分中差分估计量）提供了新的理论视角，并将其推广至存在多元结果异质性的场景。该扩展的核心优势在于允许联合结果中存在任意依赖关系与异质性。我们通过合成数据与真实数据验证其有效性。特别地，我们重新审视了Card & Krueger的经典数据集，考察最低工资上调对快餐业就业的影响；基于本方法的再分析表明，最低工资上调后，餐馆倾向于以更快速度使用兼职劳动力替代全职劳动力。本工作前一版本标题为《最优传输视角下的因果推断》。