Factorial Difference-in-Differences (FDID)

• FDID is a methodological framework that extends traditional DID by employing a 2x2 factorial design with universal exposure and a binary baseline factor.
• It distinguishes between effect modification (associative estimands) and average causal interaction (causal moderation) through explicit counterfactual definitions.
• The approach utilizes regression-based estimation with robust identification assumptions to analyze panel data where all units are exposed to a common event.

Factorial Difference-in-Differences (FDID) is a formalization of empirical strategies used in panel data contexts where all units are exposed to a common event (“universal exposure”), but heterogeneity is introduced via a baseline binary factor. Unlike canonical Difference-in-Differences (DID), FDID deploys a two-factor, $2 \times 2$ factorial design structure, clarifying the estimands, assumptions, and identification conditions underpinning such analyses. Recent formalization provides systematic estimand definitions distinguishing between associative (effect modification) and causal (causal moderation) estimands, offers generalized identification results, and specifies the precise regression-based procedures warranted in these settings (Xu et al., 2024).

1. Data Structure and Notation

FDID operates within a panel or repeated-measures dataset of $n$ units $i = 1, \ldots, n$, measured at two time points (pre and post event, $t \in \{\mathrm{pre}, \mathrm{post}\}$). The design centers on two binary factors:

• Baseline subgroup indicator $G_i \in \{0,1\}$: Classifies units into “high-$G$” ($G_i=1$) or “low-$G$” ($G_i=0$) subgroups.
• Exposure indicator $Z_i \in \{0,1\}$: Denotes whether unit $n$0 is exposed to the event ($n$1) in the post period.

In the typical FDID setting, $n$2 universally—there is no clean control group unexposed to the event; the counterfactual $n$3 is not observed but retained as a formal device for causal contrasts.

Potential outcomes are defined as $n$4: the outcome for unit $n$5 at time $n$6 under baseline status $n$7 and exposure $n$8. Observed outcomes satisfy $n$9 by the consistency assumption. The data structure allows explicit articulation of counterfactual contrasts stratified by baseline and exposure status (Xu et al., 2024).

2. Causal Estimands

FDID delineates two primary estimands within a factorial framework, distinguishing propagation and moderation of effects:

• Effect modification ($i = 1, \ldots, n$0): The associative or observed-group difference in the effect of universal exposure across levels of $i = 1, \ldots, n$1.

$i = 1, \ldots, n$2

where $i = 1, \ldots, n$3. This is descriptive, comparing average differences by $i = 1, \ldots, n$4, but not a causal effect of altering $i = 1, \ldots, n$5.

• Average causal interaction ($i = 1, \ldots, n$6): The fully causal moderation effect, defined as

$i = 1, \ldots, n$7

or, equivalently,

$i = 1, \ldots, n$8

This captures the counterfactual difference in exposure effects if the baseline status $i = 1, \ldots, n$9 were manipulated, requiring stronger identification.

These distinctions clarify that the standard DID estimator’s target depends critically on the identifying assumptions in play (Xu et al., 2024).

3. Identifying Assumptions in FDID

Five foundational assumptions structure FDID identification logic:

Assumption	Acronym	Role and Description
Universal Exposure	(UE)	$t \in \{\mathrm{pre}, \mathrm{post}\}$0 for all $t \in \{\mathrm{pre}, \mathrm{post}\}$1; all units experience the event (no unexposed group)
No Anticipation	(NA)	$t \in \{\mathrm{pre}, \mathrm{post}\}$2: Event cannot affect pre outcomes
Canonical Parallel Trends	(PT)	$t \in \{\mathrm{pre}, \mathrm{post}\}$3-groups would have parallel outcome changes post-pre in absence of exposure
Factorial Parallel Trends	(FPT)	For all $t \in \{\mathrm{pre}, \mathrm{post}\}$4, mean time-change is $t \in \{\mathrm{pre}, \mathrm{post}\}$5-independent: $t \in \{\mathrm{pre}, \mathrm{post}\}$6 equal
Exclusion Restriction	(ER)	For $t \in \{\mathrm{pre}, \mathrm{post}\}$7, $t \in \{\mathrm{pre}, \mathrm{post}\}$8: subgroup is unaffected

Assumptions (PT) are minimally sufficient for identifying $t \in \{\mathrm{pre}, \mathrm{post}\}$9 via DID. Identification of the stronger causal estimand $G_i \in \{0,1\}$0 requires (FPT). (ER) renders canonical DID a special case, with $G_i \in \{0,1\}$1 acting as a “true” control group (Xu et al., 2024).

4. Estimation and Regression Implementation

The FDID estimator extends the DID difference-of-differences:

$G_i \in \{0,1\}$2

• Under (UE), (NA), and (PT): $G_i \in \{0,1\}$3 is a consistent estimator for the effect modification $G_i \in \{0,1\}$4.
• Under (UE), (NA), (PT), and (FPT): $G_i \in \{0,1\}$5 is consistent for the average causal interaction $G_i \in \{0,1\}$6.

Regression implementation employs a two-way fixed-effects model:

$G_i \in \{0,1\}$7

Here, $G_i \in \{0,1\}$8. When incorporating baseline covariates $G_i \in \{0,1\}$9, one can include interaction and time-varying effects, centering $G$0 for interpretational clarity and valid inference. Stratified and regression-adjusted estimators enable adjustment when identifying assumptions hold only conditional on $G$1 (Xu et al., 2024).

5. Extensions and Generalizations

FDID accommodates several extensions relevant for applied research:

• Conditional and Weighted Estimation: If (PT) or (FPT) hold only conditional on $G$2, compute strata-specific DID estimators or employ regression models for $G$3. Inverse-propensity-weighting estimators (IPW-FDID) permit adjustment for $G$4-assignment probabilities under overlap, further supporting conditional parallel trends identification.
• Multiple Time Periods: For designs with multiple pre- and post-periods, pre-trends and post-trends diagnostics are recommended. Generalizations to event-study and generalized two-way fixed-effects (TWFE) specifications allow for more flexible temporal dynamics and carryover checks.
• Practical Recommendations: Explicit specification of estimand, rigorous articulation of identification assumptions, placebo/falsification checks (pre- and post-event outcomes), and correct treatment of covariates and standard errors (e.g., centering, clustering/bootstrapping) are recommended to ensure valid inference (Xu et al., 2024).

6. Canonical DID as a Special Case of FDID

The canonical two-group, two-period DID design is subsumed by FDID under the exclusion restriction (ER) for a baseline group ($G$5) that is truly unaffected by the event. In this case:

• $G$6,
• $G$7,

recovering the standard Average Treatment on the Treated (ATT) estimand, with $G$8 as a clean control group (Xu et al., 2024).

7. Implications and Distinctions

FDID formally distinguishes between descriptive/associative and causal estimands in universal-exposure panels. The choice of estimand and validity of analyst assumptions determines interpretability:

• Canonical DID estimates typically map to $G$9—effect modification, not causal moderation—unless factorial parallel trends or exclusion restrictions are precisely justified.
• Placebo and falsification strategies (pre-trends, alternative outcomes) are crucial for justification, but evidence for canonical parallel trends does not imply factorial parallel trends.
• Clearly distinguishing estimator vs. research design, and specifying the estimand and assumptions, are critical for transparency in FDID applications.

FDID thus generalizes canonical DID, providing a rigorous framework for effect heterogeneity analysis in settings with universal exposure and a baseline stratification factor, unifying a variety of empirical strategies under a coherent framework with explicit causal and associative interpretations (Xu et al., 2024).