Geodesic Synthetic Control Methods for Random Objects and Functional Data

2025-05-01

AI Summary

This paper addresses the challenge of causal inference with non-Euclidean outcomes—such as probability distributions, networks, trees, and functions—by generalizing Synthetic Control (SC) and Difference-in-Differences (DID) to arbitrary geodesic metric spaces. We propose geodesic synthetic control and geodesic DID estimators, which construct synthetic weights via optimization over geodesic distances (e.g., Wasserstein distance), ensuring double robustness. Our framework unifies and extends classical Euclidean causal inference paradigms to random objects residing in non-Euclidean spaces, providing both theoretical foundations and computationally tractable tools for distributional and structural causal analysis. The method’s validity and robustness are demonstrated through simulation studies and three empirical applications: labor market restructuring following the 2011 Tōhoku earthquake in Japan, fertility effects of East Germany’s abortion policy reform, and shifts in mortality distributions after the dissolution of the Soviet Union.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Extends synthetic control to non-scalar outcomes in metric spaces
• Generalizes methods for geodesic difference-in-differences with robustness
• Applies to complex data like distributions, networks, and functional data

Innovation

• Extends synthetic control to geodesic metric spaces
• Combines geodesic synthetic control with difference-in-differences
• Applies method to non-scalar outcomes like distributions