A Non-Bipartite Matching Framework for Difference-in-Differences with General Treatment Types

TL;DR

This paper introduces a non-bipartite matching framework for difference-in-differences that handles general treatment types without relying on parametric models or static treatment assumptions, enabling more flexible causal inference.

Contribution

It develops an optimal matching design for DID with arbitrary treatments, establishes a valid inference condition, and proposes a nonparametric causal estimand applicable to finite populations.

Findings

Balances covariates and treatment trajectories effectively

Enables valid design-based inference under new conditions

Provides a nonparametric causal measure for diverse treatments

Abstract

Difference-in-differences (DID) is one of the most widely used causal inference frameworks in observational studies. However, most existing DID methods are designed for binary treatments and cannot be readily applied to non-binary treatment settings. Although recent work has begun to extend DID to non-binary (e.g., continuous) treatments, these approaches typically require strong additional assumptions, including parametric outcome models or the presence of idealized comparison units with (nearly) static treatment levels over time (commonly called ``stayers'' or ``quasi-stayers''). In this technical note, we introduce a new non-bipartite matching framework for DID that naturally accommodates general treatment types (e.g., binary, ordinal, or continuous). Our framework makes three main contributions. First, we develop an optimal non-bipartite matching design for DID that jointly balances baseline covariates across comparable units (reducing bias) and maximizes contrasts in treatment trajectories over time (improving efficiency). Second, we establish a post-matching randomization condition, the design-based counterpart to the traditional parallel-trends assumption, which enables valid design-based inference. Third, we introduce the sample average DID ratio, a finite-population-valid and fully nonparametric causal estimand applicable to arbitrary treatment types. Our design-based approach that preserves the full treatment-dose information, avoids parametric assumptions, does not rely on the existence of stayers or quasi-stayers, and operates entirely within a finite-population framework, without appealing to hypothetical super-populations or outcome distributions.