The synthetic control method (SCM) is a widely used tool for evaluating causal effects of policy changes in panel data settings. Recent studies have extended its framework to accommodate complex outcomes that take values in metric spaces, such as distributions, functions, networks, covariance matrices, and compositional data. However, due to the lack of linear structure in general metric spaces, theoretical guarantees for estimation and inference within these extended frameworks remain underdeveloped. In this study, we propose the functional synthetic control (FSC) method as an extension of the SCM for metric space-valued outcomes. To address challenges arising from the nonlinearlity of metric spaces, we leverage isometric embeddings into Hilbert spaces. Building on this approach, we develop the FSC and augmented FSC estimators for counterfactual outcomes, with the latter being a bias-corrected version of the former. We then derive their finite-sample error bounds to establish theoretical guarantees for estimation, and construct prediction sets based on these estimators to conduct inference on causal effects. We demonstrate the usefulness of the proposed framework through simulation studies and three empirical applications.

翻译：合成控制方法（SCM）是评估面板数据中政策变化因果效应的常用工具。近期研究已将其框架扩展至适应度量空间取值的复杂结果，例如分布、函数、网络、协方差矩阵和成分数据。然而，由于一般度量空间缺乏线性结构，这些扩展框架内的估计与推断理论保证仍不完善。本研究提出功能性合成控制（FSC）方法作为SCM对度量空间值结果的扩展。针对度量空间非线性带来的挑战，我们利用等距嵌入到希尔伯特空间的技术。基于此方法，我们开发了用于反事实结果的FSC估计量及其增强版本（后者为前者的偏差校正形式）。随后推导了它们的有限样本误差界以建立估计的理论保证，并基于这些估计量构建预测集以进行因果效应推断。通过模拟研究和三个实证应用，我们证明了所提框架的有效性。