Not So Harmless After All: The Fixed-Effects Model

The Conventional Case for Fixed Effects

The FE estimator controls for time-invariant unobserved heterogeneity by removing between-unit variation (the “within” transformation):

\tilde{y}_{it} = \beta\,\tilde{x}_{it} + \tilde{\varepsilon}_{it}

where \tilde{z}_{it} = z_{it} - \bar{z}_i.

Key virtue: if unit effects u_i are correlated with regressors, OLS is biased but FE is unbiased — provided the model is otherwise correctly specified.

“Under a strict exogeneity assumption on the explanatory variables, the fixed-effects estimator is unbiased.” — Wooldridge (2002)

The Catch: Strict Exogeneity

Strict exogeneity requires that no time-varying covariate is omitted and that dynamics are correctly modelled.

In political science panels this assumption is frequently violated because:

• Time-varying confounders are common and hard to measure
• Dynamic misspecifications are pervasive — most papers use ad hoc fixes (lagged DV, period dummies) rather than the true lag structure
• The correct lag length is rarely identifiable from the data

P&T’s claim: violating strict exogeneity hurts FE more than it hurts OLS.

Two Distinct Sources of Bias

The problem: FE solves the first issue at the cost of concentrating all identification in within variation, making it maximally exposed to the second issue.

Formal Result 1: Bias with an Omitted Time-Varying Variable

True DGP: y_{it} = \alpha_1 x_{it} + \alpha_2 w_{it} + u_i + \varepsilon_{it}

Omit w_{it}, where w_{it} = \gamma_1 \ddot{x}_{it} + \gamma_2 \bar{x}_i + \omega_{it}
(\gamma_1 = correlation with within variation; \gamma_2 = correlation with between variation)

\text{Bias}(\hat{\alpha}_{1,\text{OLS}}) = \frac{\gamma_1 \sum\ddot{x}^2 + T\gamma_2\sum\bar{x}^2 + \delta_1 T\sum\bar{x}^2}{\sum\ddot{x}^2 + T\sum\bar{x}^2}

\text{Bias}(\hat{\alpha}_{1,\text{FE}}) = \gamma_1

FE bias equals \gamma_1 regardless of \delta_1 (correlation with unit effects). FE is more biased than OLS whenever the within correlation \gamma_1 dominates.

Formal Result 2: Bias from a Misspecified Lag Structure

True DGP: y_{it} = \beta x_{it-1} + u_i + \varepsilon_{it}
Estimated (incorrectly) as: y_{it} = \beta x_{it} + \varepsilon_{it}

The probability limit of the FE estimator is:

\text{plim}(\hat{\beta}_{\text{FE}}) = \beta\frac{\text{Cov}(x_{it-1}, x_{it})}{\text{Var}(x_{it})} - \beta\frac{\text{Var}(\bar{x}_i)}{\text{Var}(x_{it})}

The second term is unique to FE: discarding between variation subtracts additional bias. When x is positively autocorrelated (the typical case), FE bias exceeds OLS bias.

The General Mechanism

Biases from different misspecifications are not additive.

FE solves the time-invariant omitted-variable problem by discarding between variation. But this concentrates estimation entirely in within variation — making dynamic misspecification more, not less, damaging.

• The “fix” for one problem worsens another.
• In realistic settings where both time-invariant and time-varying omissions exist, neither estimator dominates, but FE performs poorly when within correlation is large.

Key Result 1: Omitted Time-Varying Variable

• FE bias increases with \gamma_1 (within correlation) and is unaffected by the correlation with u_i.
• OLS bias is a weighted combination of within and between correlations; it is smaller than FE when \gamma_1 is large.
• Common econometric patches (lagged DV, period dummies, Prais–Winsten) do not reduce FE bias in this setting.
• Overall, pooled-OLS produces up to 45% less bias than FE across experiments.

Key Result 2: Omitted Trends

When both x^1 and an omitted variable share a common trend:

• FE eliminates between variation but trends survive the within transformation → FE is still biased.
• OLS absorbs part of the trend bias through between variation → lower total bias in many cells.

When trends are unit-specific (heterogeneous), the problem intensifies for FE: each unit’s trend component remains in the within variation but now also pollutes FE in ways that partial fixes cannot address.

Key Result 3: Misspecified Lag Structure

When the true effect operates with a one-period lag (or heterogeneous lags across units):

• FE bias is downward and larger in absolute value than OLS bias whenever x is positively autocorrelated.
• Bias grows with T for FE (more within variation accumulated, all contaminated by the wrong lag).
• ADL(1,1) substantially reduces bias in both OLS and FE, but FE still suffers more from the residual misspecification.

The Hausman Test Problem

The Hausman test is commonly used to choose between FE and RE/OLS:

If FE and OLS estimates differ significantly → use FE (it is “consistent”)

P&T show this logic fails under multiple misspecifications:

• When a time-varying omitted variable exists alongside unit effects, the Hausman test erratically recommends FE even when OLS is less biased.
• Including a lagged DV biases the Hausman test toward FE.
• With no omitted time-invariant variable but dynamic misspecification, the test still frequently rejects in favour of FE — the wrong choice.

Conclusion: the Hausman test is not a reliable guide to model selection when the world contains both time-varying and time-invariant problems simultaneously.

Summary of Findings

Implications for Applied Researchers

What P&T are not saying: Pooled-OLS is not a recommended alternative — it has its own well-known biases.

What they are saying:

• Do not use FE as a default without justifying that dynamics are correctly specified.
• Model dynamics first, before worrying about time-invariant unobservables.
• Be sceptical of the Hausman test as a sole basis for model selection.
• Consider robustness across dynamic specifications (e.g. distributed lag models, ADL) rather than committing to one.
• Report evidence on serial correlation in residuals — its absence is a prerequisite for trusting FE.

Broader Methodological Point

Econometric solutions designed to fix a single misspecification can exacerbate bias from a different misspecification.

This challenges the standard practitioner heuristic of applying modular fixes sequentially (add FE → add LDV → done). Multiple simultaneous misspecifications interact in non-additive ways.

P&T call on methodologists to study estimator properties under combined misspecifications — not one at a time.

Reference

Plümper, Thomas and Vera E. Troeger. 2019.
“Not So Harmless After All: The Fixed-Effects Model.”
Political Analysis 27(1): 21–45.
DOI: 10.1017/pan.2018.17

Keywords: consistency · efficiency · misspecification · omitted variable bias · panel data · Monte Carlo simulation

Panel Data: Two Dimensions of Variation

For unit i at time t, observe (Y_{it},\, X_{it}).

One-Way Fixed Effects

Case (unit) FE: Y_{it} = \alpha_i + \beta X_{it} + \varepsilon_{it}

• Subtracts each unit’s mean → estimates within-unit, over-time variation
• \hat\beta_{\text{unit FE}} = variance-weighted average of unit-specific slopes

Time FE: Y_{it} = \alpha_t + \beta X_{it} + \varepsilon_{it}

• Subtracts each period’s mean → estimates cross-sectional variation
• \hat\beta_{\text{time FE}} = variance-weighted average of period-specific slopes

Difference-in-Differences (DiD) — 2 × 2 Case

Two groups (treated s, never-treated n) × two periods (pre/post):

\widehat{\text{DiD}} = \bigl(\bar Y_{s,2} - \bar Y_{s,1}\bigr) - \bigl(\bar Y_{n,2} - \bar Y_{n,1}\bigr)

Identifying assumption — Parallel Trends: \mathbb{E}[Y_{s,2}(0) - Y_{s,1}(0)] = \mathbb{E}[Y_{n,2}(0) - Y_{n,1}(0)]

Under parallel trends, DiD identifies the Average Treatment Effect on the Treated (ATT).

Two-Way Fixed Effects (TWFE)

Y_{it} = \alpha_i + \lambda_t + \beta D_{it} + \varepsilon_{it}

• \alpha_i: unit fixed effects (absorb time-invariant heterogeneity)
  
• \lambda_t: time fixed effects (absorb common time shocks)
  
• D_{it}: treatment indicator (or intensity)

Standard interpretation: \beta is “the causal effect of treatment, controlling for unit- and time-level confounders.”

Kropko & Kubinec (2020): What Does TWFE Actually Estimate?

PLoS ONE 15(4): e0231349

TWFE: A Complex Amalgamation

Kropko & Kubinec simulate TSCS data with known within-case slope \gamma and within-time slope \beta:

The Identification Problem

• The design matrix is rank-deficient; software (R, Stata) silently drops a dummy, masking the problem.
• Near-constant slopes → coefficient variance explodes.

Why has this gone unnoticed?
Automatic multicollinearity handling in lm() and xtreg suppresses the error; results look plausible but are essentially arbitrary.

When TWFE = DiD (and When It Doesn’t)

Kropko & Kubinec (2020): TWFE should be thought of as a generalised DiD only in the canonical 2 × 2 design.

Staggered DiD & Heterogeneous Effects

Recent literature (de Chaisemartin & D’Haultfœuille 2020, 2022; Callaway & Sant’Anna 2021; Goodman-Bacon 2021) shows:

\hat\beta_{\text{TWFE}} = \sum_{g,t} w_{g,t} \cdot \text{ATT}(g,t)

where some weights w_{g,t} can be negative — earlier treated units act as “controls” for later treated units.

Intuition: The Negative-Weight Problem

Time →        t=1    t=2    t=3    t=4
Group A       0      1      1      1   ← treated early
Group B       0      0      1      1   ← treated mid
Group C       0      0      0      0   ← never treated

TWFE uses Group A (post-treatment) as a control for Group B in periods 3–4.
If Group A’s treatment effect grows over time, this biases \hat\beta_{\text{TWFE}} downward.

Recommendations

For staggered DiD, use robust estimators: This is Friday

Summary

Panel data question
       │
       ├─ Over-time (within-unit) → Unit FE
       │
       ├─ Cross-sectional        → Time FE
       │
       └─ Causal treatment effect?
              │
              ├─ 2 × 2, parallel trends → TWFE = DiD ✅
              │
              └─ Staggered / long panel → Use modern DiD estimators ✅
                                          Avoid plain TWFE ❌

Key References

• Kropko J, Kubinec R (2020). “Interpretation and identification of within-unit and cross-sectional variation in panel data models.” PLoS ONE 15(4): e0231349.
• de Chaisemartin C, D’Haultfœuille X (2020). “Two-way fixed effects estimators with heterogeneous treatment effects.” American Economic Review 110(9): 2964–2996.
• Goodman-Bacon A (2021). “Difference-in-differences with variation in treatment timing.” Journal of Econometrics 225(2): 254–277.
• Callaway B, Sant’Anna PHC (2021). “Difference-in-differences with multiple time periods.” Journal of Econometrics 225(2): 200–230.
• Sun L, Abraham S (2021). “Estimating dynamic treatment effects in event studies with heterogeneous treatment effects.” Journal of Econometrics 225(2): 175–199.