Abstract

The synthetic control method (SCM) is a widely used tool for evaluating the causal effects of policy changes in panel data settings. Recently, several studies have extended this framework to accommodate complex outcomes that take values in metric spaces, such as distributions, functions, networks, correlation matrices, and compositional data. However, theoretical guarantees for estimation and inference within these extended frameworks remain underdeveloped. In this study, we propose a novel extension of the SCM for metric space-valued outcomes. To address challenges arising from the lack of linear structure in general metric spaces, we leverage isometric embeddings of metric spaces into Hilbert spaces. Building on this approach, we develop estimators for missing potential outcomes, derive their finite-sample error bounds, and construct prediction sets for causal effects. The proposed methods are demonstrated through simulations and empirical applications.