Factorial Difference-in-Differences (2407.11937v3)

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.

• The paper unifies canonical DID with FDID by demonstrating that traditional DID is a special case under an exclusion restriction assumption.
• It introduces a factorial design that separately identifies effect modification and causal interaction through baseline factors and exposure levels.
• The methodology extends to incorporate covariate adjustments and multiple time periods, enhancing robust causal inference in universally exposed settings.

An Overview of Factorial Difference-in-Differences

The paper "Factorial Difference-in-Differences" provides an in-depth exploration of the factorial difference-in-differences (FDID) research design, a variant of the traditional difference-in-differences (DID) methodology widely used in social science for causal inference. This research is primarily concerned with causal analysis in settings where a clean control group is not present, a scenario not uncommon in social science applications where all units are exposed to an event. The paper aims to clarify the capabilities of FDID in extracting causal effects through a nuanced understanding of data structure, estimands, and identifying assumptions.

The Factorial DID Framework

FDID is set apart from standard DID by the absence of a conventional control group unexposed to the event in question post-exposure. Instead, FDID is conceptualized via a factorial design with two components: a baseline factor $G$, and the exposure level $Z$ induced by the event. This distinction permits the isolation of two primary effects: effect modification and causal interaction. Effect modification refers to the associative impact of $G$ on the effect of $Z$, whereas causal interaction pertains to causal modulation by $G$ on the exposure impact $Z$.

In the classical DID framework, causal identification typically hinges upon the no anticipation and parallel trends assumptions. FDID extends this by maintaining the former while introducing an additional factorial parallel trends assumption. This addition seeks to extract causal interaction—a feat not achievable with the canonical parallel trends assumption alone.

Key Contributions

One of the primary contributions of this work is the theoretical unification of canonical DID and FDID. The paper demonstrates that traditional DID can be viewed as a special instance of FDID by applying an exclusion restriction assumption which asserts non-effect of the event on units with a certain baseline factor level. This perspective broadens the methodological toolkit for researchers, simultaneously extending applicability and interpretability beyond settings with clear control groups.

Additionally, the research offers extensions to include conditioning on covariates and accommodating multiple time periods. It incorporates regression analysis and inverse propensity weighting to account for covariate imbalance, promoting robust identification of causal effects in settings eluding straightforward control comparisons.

Implications and Applications

The FDID framework not only fills an unmet need in studying settings impacted ubiquitously by historical events but also refines our understanding of effect heterogeneity in complex social phenomena. It holds considerable potential for unveiling interactions in observational data, where baseline factors have a modifying impact on exposure effects. This could be transformative in evaluating policies or historical economic shifts, where clean control groups are often nonexistent.

Applying FDID may necessitate nuanced examinations of underlying assumptions to justify causal interpretations. This poses a challenge, urging researchers to consider auxiliary evidence, such as placebo analyses or in-depth discussions on assumption plausibility.

Looking Forward

Future research directions include extending the FDID model to incorporate continuous or multinomial baseline factors and bridging further statistical methodologies with FDID, such as machine learning models for richer, more nuanced causal inference frameworks. This methodological advancement opens the way for extensive empirical applications, borrowing from the robustness of factorial experimental designs to strengthen causal inferences in observational studies across various disciplines.

In conclusion, the paper advances a critical dialogue in causal inference, offering both theoretical augmentation and practical applicability of the DID estimator in settings characterized by universal treatment exposure. The proposed FDID approach underscores a departure from traditional emphasis solely on treatment effects in favor of nuanced interaction effects, offering a lens to decode complex causal architectures in socio-political and economic domains.