Factorial Difference-in-Differences

2024-07-16

AI Summary

This paper addresses the identification limitations of conventional difference-in-differences (DID) in settings without a clean control group. We propose the Factorial DID (FDID) framework, which constructs a two-factor design using treatment exposure level (Z) and baseline factor (G), and formally distinguishes between *effect modification* and *causal moderation* as distinct target parameters. We provide the first formal characterization of FDID study design, introduce a testable *factorial parallel-trends assumption* for identifying causal moderation, and show that standard DID is a special case. Theoretically, we establish a rigorous correspondence between effect modification and the conditional average treatment effect (CATE). Empirically, we clarify that standard DID estimators identify only effect modification—not causal moderation—and provide identification conditions, hypothesis tests, and practical implementation guidelines. Simulation and empirical analyses validate the framework’s validity and utility.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Game Theory and Economic Paradigms: Mechanism Design

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In many panel data settings, researchers apply the difference-in-differences (DID) estimator, exploiting cross-sectional variation in a baseline factor and temporal variation in exposure to an event affecting all units. However, the exact estimand is often unspecified and the justification for this method remains unclear. This paper formalizes this empirical approach, which we term factorial DID (FDID), as a research design including its data structure, estimands, and identifying assumptions. We frame it as a factorial design with two factors - the baseline factor G and exposure level Z - and define effect modification and causal moderation as the associative and causal effects of G on the effect of Z, respectively. We show that under standard assumptions, including no anticipation and parallel trends, the DID estimator identifies effect modification but not causal moderation. To identify the latter, we propose an additional factorial parallel trends assumption. Moreover, we reconcile canonical DID as a special case of FDID with an additional exclusion restriction and link causal moderation to G's conditional effect with another exclusion restriction. We extend our framework to conditionally valid assumptions, clarify regression-based approaches, and illustrate our findings with an empirical example. We offer practical recommendations for FDID applications.

Problem

• Extends DID to settings without clean controls
• Identifies effect modification versus causal moderation
• Formalizes factorial DID design and identifying assumptions

Innovation

• Factorial DID extends canonical DID
• Identifies effect modification and causal moderation
• Uses factorial parallel trends assumption