Two-Way Fixed Effects, the Two-Way Mundlak Regression, and DiD Estimators
Causal Question / Estimand
Average treatment effects on the treated, including cohort- and period-specific effects and event-study dynamics (Causal-Estimand), recovered from flexible regression rather than abandoning fixed effects.
Identification Strategy
Proves an algebraic equivalence: the TWFE estimator equals a pooled-OLS “two-way Mundlak” regression that includes unit-specific and time-specific averages. Building on this, an extended TWFE (ETWFE) — pooled OLS that fully interacts treatment-cohort indicators, period indicators, and covariates — allows rich Treatment-Effect-Heterogeneity and is shown to equal an imputation estimator derived under no anticipation and parallel trends. The lesson: TWFE is not inherently broken; unsaturated TWFE is. Saturating the regression identifies the ATTs.
Key Assumptions
Parallel-Trends (including cohort-specific trends in extended versions), No-Anticipation, correct (flexible) functional form for cohort/period/covariate interactions, and SUTVA.
Threats to Validity
n/a — methodological/algebraic. Caveat: the equivalence and unbiasedness rely on correctly specifying the interactions; under-parameterizing reintroduces the heterogeneity bias / Negative-Weighting of plain TWFE.
Setting / Data
n/a — econometric methodology. Connects to the etwfe/Mundlak implementations;
illustrated with staggered-intervention panel examples.
Key Claims
- TWFE ≡ two-way Mundlak pooled OLS — a unifying algebraic result.
- Extended TWFE (fully interacted POLS) ≡ an imputation estimator and identifies ATTs under PT and no anticipation, allowing heterogeneity.
- Event-study versions permit pre-trend testing and cohort-specific trends.
Connections
- Robust-estimator family alongside: CallawaySantAnna2021-DiDMultiplePeriods, SunAbraham2021-EventStudies, DeChaisemartinDHaultfoeuille2020-IntertemporalTE
- Reframes the diagnosis of: GoodmanBacon2021-DiDVariationInTiming
- Companion Mundlak result: ArkhangelskyImbens2024-FixedEffectsGeneralizedMundlak — same algebraic lever, but redesigns the estimator around the assignment process (grouped cross-sections) rather than saturating the regression
- Imputation sibling: BorusyakEtAl2024-RevisitingEventStudyDesigns — same fit-on-untreated-then-impute logic, framed as the efficient robust estimator
- See also: DeChaisemartinDHaultfoeuille2023-TWFESurvey, DiD
Citation
Wooldridge, J. M. (2021). Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimators. Working paper. Published (2025) in Empirical Economics, 69, 2545–2587. https://doi.org/10.1007/s00181-025-02807-z