Geodesic Difference-in-Differences

TL;DR

This paper introduces geodesic DID, a new causal inference framework for outcomes in non-Euclidean spaces like distributions and networks, addressing the limitations of traditional DID methods.

Contribution

It develops a novel geodesic DID approach for metric space outcomes, establishing identification, convergence, and extending to staggered treatment settings.

Findings

Analyzed health impacts of the Soviet Union's collapse using age-at-death distributions.

Assessed effects of U.S. electricity market liberalization on generation composition.

Demonstrated the applicability of geodesic DID to complex, non-Euclidean data.

Abstract

Difference-in-differences (DID) is a widely used quasi-experimental design for causal inference, traditionally applied to scalar or Euclidean outcomes, while extensions to outcomes residing in non-Euclidean spaces remain limited. Existing methods for such outcomes have primarily focused on univariate distributions, leveraging linear operations in the space of quantile functions, but these approaches cannot be directly extended to outcomes in general metric spaces. In this paper, we propose geodesic DID, a novel DID framework for outcomes in geodesic metric spaces, such as distributions, networks, and manifold-valued data. To address the absence of algebraic operations in these spaces, we use geodesics as proxies for differences and introduce the geodesic average treatment effect on the treated (ATT) as the causal estimand. We establish the identification of the geodesic ATT and derive the convergence rate of its sample versions, employing tools from metric geometry and empirical process theory. This framework is further extended to the case of staggered DID settings, allowing for multiple time periods and varying treatment timings. To illustrate the practical utility of geodesic DID, we analyze health impacts of the Soviet Union's collapse using age-at-death distributions and assess effects of U.S. electricity market liberalization on electricity generation compositions.