We introduce a geodesic synthetic control method for causal inference that extends existing synthetic control methods to scenarios where outcomes are elements in a geodesic metric space rather than scalars. Examples of such outcomes include distributions, compositions, networks, trees and functional data, among other data types that can be viewed as elements of a geodesic metric space given a suitable metric. We extend this further to geodesic synthetic difference-in-differences that builds on the established synthetic difference-in-differences for Euclidean outcomes. This estimator generalizes both the geodesic synthetic control method and a previously proposed geodesic difference-in-differences method and exhibits a double robustness property. The proposed geodesic synthetic control method is illustrated through comprehensive simulation studies and applications to the employment composition changes following the 2011 Great East Japan Earthquake, and the impact of abortion liberalization policy on fertility patterns in East Germany. We illustrate the proposed geodesic synthetic difference-in-differences by studying the consequences of the Soviet Union's collapse on age-at-death distributions for males and females.

翻译：我们提出了一种用于因果推断的测地线合成控制方法，将现有的合成控制方法扩展到结果变量为测地度量空间中的元素而非标量的场景。此类结果变量的示例包括分布、成分、网络、树和函数数据等，这些数据类型在给定适当度量后可被视为测地度量空间中的元素。我们进一步将其扩展为测地线合成双重差分法，该方法建立在成熟的欧几里得结果变量合成双重差分法基础上。该估计器同时推广了测地线合成控制方法和先前提出的测地线双重差分方法，并展现出双重稳健性。所提出的测地线合成控制方法通过全面的模拟研究以及应用于2011年东日本大地震后就业结构变化和堕胎自由化政策对东德生育模式影响的实证分析得到验证。我们通过研究苏联解体对男性和女性死亡年龄分布的影响，展示了所提出的测地线合成双重差分法的应用。