Fixed Effects and the Generalized Mundlak Estimator

Causal Question / Estimand

Average treatment effects (Causal-Estimand) in observational studies with unobserved group-level heterogeneity (units nested in states, cities, classrooms, families…). Shows that under general Treatment-Effect-Heterogeneity the population ATE is not point-identified from grouped data; the fixed-effect estimator targets an unusual weighted average whose weights depend on the sampling scheme (e.g. its meaning changes if you sample three units per group instead of two), and with covariates plain FE may estimate no meaningful weighted average at all.

Identification Strategy

Unpacks the fixed-effect regression into three separable assumptions (constant effects, group unconfoundedness, linearity/additivity), then exploits Mundlak’s (1978) equivalence: the FE estimator equals a regression without fixed effects that controls for group averages of treatment and covariates. Group unconfoundedness implies unconfoundedness conditional on group-level balancing scores — $W_i\perp (Y_i(0),Y_i(1))\mid X_i,{\bar{H}}_{g(i)}$ — which validates some between-group comparisons and gives groups a similarity metric that raw group indicators lack. This motivates Generalized Mundlak Estimators (GMEs): flexible (non-linear) adjustment for group averages, propensity-score weighting on unit and group characteristics, and doubly robust combinations — importing the modern unconfoundedness toolkit into settings with small, fixed group sizes where the incidental-parameters problem rules out within-group estimation.

Key Assumptions

Ignorability (group unconfoundedness — assignment ignorable given group membership and covariates, strengthened to conditioning on group balancing scores), Propensity-Score (balancing-score logic at the group level), Overlap (propensity trimming to units with score in $[0.05,0.95]$), SUTVA (no spillovers across groups — flagged as restrictive in the blood-drive application), and random two-stage sampling of groups then units.

Threats to Validity

Largely theoretical, but the paper itself is a threat catalogue for applied FE work: under heterogeneity, FE estimands are sampling-scheme-dependent conditional averages (e.g. $\tau (1/2)$, not $\tau$); with covariates, unsaturated FE combines validated and non-validated cross-group comparisons. For GMEs: misspecification of the chosen balancing scores $H(W,X)$, and cross-group interference (blood supply in neighboring locations is interdependent).

Setting / Data

Empirical illustration: American Red Cross blood drives (Lacetera et al.) — effect of economic incentives on blood units collected; 2,491 host locations (groups), 14,029 drives (units). FE estimate 4.12 vs doubly robust GME 3.42. Data-driven Monte Carlo (zero-effect placebo): GME beats linear FE on bias (0.38 vs 0.79) and RMSE.

Key Claims

• Group unconfoundedness implies unconfoundedness conditional on group-level balancing scores (averages of functions of treatment and covariates), which justifies some between-group comparisons and defines similarity between groups.
• Under treatment-effect heterogeneity the FE estimator targets a weighted average whose weights depend on the sampling and assignment scheme; with covariates it can fail to estimate any weighted average effect.
• GMEs — flexible outcome modelling + estimated propensity scores + doubly robust combination on $(X_i,{\bar{S}}_{g(i)})$ — recover well-defined ATEs with small, fixed group sizes, with proven consistency and asymptotic normality.
• Empirically and in simulation the doubly robust GME outperforms linear FE.

Connections

• Builds on: Rubin1974-EstimatingCausalEffects (potential outcomes), Imbens2004-NonparametricATEReview and AbadieCattaneo2018-EconometricMethodsProgramEvaluation (the modern unconfoundedness/doubly-robust toolkit it imports)
• Companion result to: Wooldridge2021-TWFEMundlakDiD — both exploit the Mundlak equivalence; Wooldridge saturates the regression, Arkhangelsky–Imbens redesign the estimator around the assignment process
• Same diagnosis, different setting: GoodmanBacon2021-DiDVariationInTiming, DeChaisemartinDHaultfoeuille2020-IntertemporalTE — FE under Treatment-Effect-Heterogeneity yields hard-to-interpret weighted averages
• Doubly robust family: SantAnnaZhao2020-DoublyRobustDiD
• Design-based kinship (identification from the assignment process, not outcomes): AbadieEtAl2020-SamplingVsDesignUncertainty, AtheyImbens2022-DesignBasedDiD
• See also: DiD, PSM

Citation

Arkhangelsky, D., & Imbens, G. W. (2024). Fixed Effects and the Generalized Mundlak Estimator. Review of Economic Studies, 91(5), 2545–2571. https://doi.org/10.1093/restud/rdad089