Fixed and Random Effects Models: Making an Informed Choice

Robert W. Walker

2026-07-27

Overview

“In most research scenarios, a well-specified RE model provides everything that FE provides and more.”

This paper:

Clarifies widespread confusion about fixed effects (FE) and random effects (RE) models
Introduces the within-between RE (REWB) model as the most general framework
Uses simulations to evaluate behaviour under likely mis-specifications
Provides practical guidance for applied researchers

The Problem in the Literature

Two camps with contradictory advice:

Camp	View
RE advocates	RE provides more information and is superior in most settings
FE advocates	FE is the “gold standard” — protects against endogeneity

Root causes of confusion:

Inconsistent terminology across disciplines (especially “random effects”)
Misreading the Hausman test as a simple FE-vs-RE selector
Failure to distinguish within from between variation

Data Structures: Setting the Scene

Social science data are typically multilevel:

Structure	Level 1	Level 2	Level 3
Cross-sectional	Individuals	Neighbourhoods	—
Cross-national	Individuals	Countries	—
Panel (individuals)	Occasions	Individuals	—
Panel (countries)	Occasions	Countries	—
Three-level	Individuals	Country-years	Countries

Crucially: relationships can differ within and between clusters — these should not be assumed equal.

Within vs Between Effects

Consider income (x) predicting some outcome (y):

Within effect (level 1): How does a person’s outcome change when their own income changes over time?
Between effect (level 2): Do people with higher average incomes tend to have higher outcomes?

Classic example (Gelman 2008):

Between states: Higher-income US states → more Democratic
Within states: Higher-income voters → more Republican

These effects can point in opposite directions. Combining them is misleading.

The Modelling Hierarchy

Most general
      │
      ▼
┌─────────────────────────────────────────┐
│  REWB / Mundlak model                   │  ← Recommended starting point
│  (separate within & between effects)    │
└─────────────────────────────────────────┘
      │  constrain β_W = β_B
      ▼
┌─────────────────────────────────────────┐
│  Standard RE model                      │  ← OK if β_W ≈ β_B
└─────────────────────────────────────────┘
      │  remove between-level info
      ▼
┌─────────────────────────────────────────┐
│  Fixed Effects (FE) model               │  ← Loses between effects entirely
└─────────────────────────────────────────┘
      │  ignore clustering
      ▼
┌─────────────────────────────────────────┐
│  Pooled OLS                             │  ← Generally inadvisable
└─────────────────────────────────────────┘
Least general / most restricted

The REWB Model (Core Specification)

y_{it} = \beta_0 + \beta_{1W}(x_{it} - \bar{x}_i) + \beta_{2B}\bar{x}_i + \beta_3 z_i + (v_i + \epsilon_{it})

Term	Meaning
\beta_{1W}	Within effect of x — variation around individual mean
\beta_{2B}	Between effect of x — individual mean level
\beta_3	Effect of time-invariant variable z_i
v_i	Random intercept for individual i

The Mundlak formulation (1978) is an alternative parameterisation with identical fit and identical within estimates.

Relationship: \beta_{1W} + \beta_{2C} = \beta_{2B}

REWB and Mundlak: Overwhelming Commonality

Both models are mathematically equivalent — same log-likelihood, same fit, same within effect:

REWB (within-between)

y_{it} = \beta_0 + \beta_{1W}\underbrace{(x_{it} - \bar{x}_i)}_{\text{demeaned}} + \newline \beta_{2B}\underbrace{\bar{x}_i}_{\text{group mean}} + \beta_3 z_i + v_i + \epsilon_{it}

x_{it} split into two orthogonal components
\bar{x}_i and (x_{it}-\bar{x}_i) are uncorrelated by construction
Reports a between effect \beta_{2B}

Mundlak (correlated RE)

y_{it} = \beta_0 + \beta_{1W}\underbrace{x_{it}}_{\text{raw}} + \newline \beta_{2C}\underbrace{\bar{x}_i}_{\text{group mean}} + \beta_4 z_i + v_i + \epsilon_{it}

x_{it} enters in raw form; \bar{x}_i added as a control
Reports a contextual effect \beta_{2C}
Collinearity between x_{it} and \bar{x}_i possible if many predictors

\boxed{\hat{\beta}_{1W}^{\text{REWB}} = \hat{\beta}_{1W}^{\text{Mundlak}}} \qquad \hat{\beta}_{2B} = \hat{\beta}_{1W} + \hat{\beta}_{2C}

REWB vs Mundlak: The Subtle Difference

The only substantive difference is what the higher-level coefficient means:

	REWB: \hat{\beta}_{2B}	Mundlak: \hat{\beta}_{2C}
What it asks	“What is the association with \bar{x}_i alone?”	“What does the context add, given individual x_{it}?”
Controls for	Nothing held constant at level 1	Individual-level x_{it} held constant
Longitudinal data	✓ Preferred — level 1 units can’t move between level 2 clusters	✗ Contextual interpretation is awkward
Cross-sectional data	✓ Valid	✓ Preferred — individual moves between contexts is meaningful

Practical takeaway:

For panel data → use REWB (between effect is interpretable; contextual effect is not)
For cross-sectional multilevel data → Mundlak contextual effect answers the policy-relevant question
Either way: fit one, recover the other by arithmetic
Both are preferable to standard RE or FE in almost all circumstances

Standard RE Model: A Constrained REWB

y_{it} = \beta_0 + \beta_1^{RE} x_{it} + \beta_3^{RE} z_i + (v_i + \epsilon_{it})

This implicitly assumes \beta_{1W} = \beta_{2B} (within = between).

When this holds: RE is more efficient than REWB — it uses all variation.

When this fails: \beta_1^{RE} is a weighted average of two different processes — substantively uninterpretable.

Test: Use a Wald test of the contextual effect in the Mundlak model, or test \beta_{1W} = \beta_{2B} in REWB. This is what the Hausman test actually does — not a simple FE vs RE selector.

Fixed Effects Model: What It Does and Doesn’t Do

y_{it} = \beta_1(x_{it} - \bar{x}_i) + (v_i + \epsilon_{it})

Equivalent to including a dummy variable for every level-2 entity.

Strengths

Within effect unbiased by between-level omitted variables
No distributional assumptions on v_i

Weaknesses

Cannot estimate between effects
Cannot include time-invariant variables (z_i)
Cannot model random slopes
Cannot add further hierarchical levels
“Throws away important information” (Nerlove 2005)

Key insight: REWB gives identical within-effect estimates to FE, plus the between effect and more.

The Hausman Test: Widely Misunderstood

Common (mis)use:

Hausman test significant → use FE; not significant → use RE

What it actually tests:

Are the within and between effects equal?
i.e., is the contextual effect \beta_{2C} = 0?

Better approach:

Fit REWB and inspect the within and between effects separately
A significant difference means the effects are substantively distinct — both are worth knowing
Use REWB regardless; it solves the problem the Hausman test detects

Omitted Variable Bias

	Level 1 OVB	Level 2 OVB
FE / REWB	✗ Not controlled	✓ Absorbed into between effect
Standard RE	✗ Not controlled	✗ Can bias within estimate
OLS	✗ Not controlled	✗ Biases all estimates

For within effects: REWB and FE equally protect against time-invariant omitted variables.

For between effects: Only REWB estimates them — and these estimates should be interpreted as proxies for broader unmeasured social processes, not direct causal effects.

Random Slopes: A Critical Extension

So far: assumed the within effect \beta_{1W} is the same for all level-2 entities.

Often wrong. The full REWB model adds:

y_{it} = \beta_0 + \beta_{1W}(x_{it}-\bar{x}_i) + \beta_{2B}\bar{x}_i + \beta_3 z_i + v_{i0} + v_{i1}(x_{it}-\bar{x}_i) + \epsilon_{it}

where v_{i1} captures heterogeneity in the within effect across individuals/countries/groups.

\sigma_{v01}	Interpretation
Negative	“Fanning in” — effects converge as x increases
Positive	“Fanning out” — effects diverge as x increases

Why Random Slopes Matter: Simulation Evidence

Finding 1: When random slopes exist in the data but are omitted from the model:

Standard errors are anti-conservative (underestimated)
Applies to FE and random-intercept RE models
Robust SEs help but do not fully correct the problem

Finding 2: Two influential reanalyses (Table 2):

Study	Without RS	With RS
Milner & Kubota (2005): Democracy → trade	β = −0.227**	β = −0.143 (NS)
Reinhart & Rogoff (2010): Debt → growth	SE triples; effect insignificant when time controlled

Recommendation: At minimum, test whether the key predictor’s coefficient changes when allowed to vary randomly.

Three-Level and Cross-Classified Models

A further advantage of REWB/RE over FE:

FE at level 2 exhausts degrees of freedom — cannot accommodate a third level
RE framework handles three or more levels parsimoniously
- One variance parameter per level
- Variance at each level indicates how much that scale “matters”

Example:

\text{Individuals} \subset \text{Country-years} \subset \text{Countries}

Cross-national surveys fielded repeatedly — common in comparative social science.

FE cannot model this. REWB can.

Non-Continuous Outcomes (GLMs)

For binary/count outcomes (logit, probit, Poisson):

REWB with non-identity link does not perfectly partition within and between effects
Bias can arise when \bar{x}_i and v_i are non-linearly related

However:

Brumback et al. (2010): “difficult to find an example where the problem is severe”
Only problematic when \bar{x}_i and v_i are highly correlated with few obs per cluster
Many link functions (ordered logit, probit, negative binomial) have no conditional likelihood alternative

Practical check: Compare REWB estimate to the conditional likelihood estimate. If similar, REWB results are reliable.

Robustness of the Normality Assumption

RE models assume v_i \sim N(0, \sigma^2_v). How much does this matter?

For linear (continuous) models — reassuring findings:

\hat{\beta} and standard errors: unbiased even under Chi-square or outlier distributions
Level-2 variance estimates: slightly biased when distribution is skewed
Random intercept estimates: biased mainly when outliers are present (but outliers can be “dummied out”)

For non-linear models — more caution needed:

Significant bias in variance estimates when level-2 variance is large
Beta estimates still generally robust

Alternatives if concerned: Non-parametric maximum likelihood (NPML) methods.

Practical Recommendations

1. Use REWB as your starting point

More general than FE or standard RE
Retains between effects; allows random slopes and additional levels
Small biases from Normality violations are usually worth the added flexibility

2. Test for random slopes

Check whether the key predictor’s effect varies across clusters
Use robust SEs as a diagnostic (large difference → possible random slope mis-specification)
Be parsimonious: not all variables need random slopes

3. Reframe the Hausman test

Don’t use it to choose between FE and RE
Use it to assess whether within and between effects differ
If they differ, model them separately with REWB — both are substantively interesting

When Is FE Acceptable?

The authors identify only three situations where FE may be preferred:

Higher-level variables are of no interest whatsoever — but this is rarely true in social science
No random slopes in the true DGP — but testing this requires fitting RE anyway
Very few level-2 entities — making random-effects estimation unreliable

In practice, REWB is almost always the better choice. Labelling REWB as a “hybrid FE” model (as some do) undersells and misrepresents its value.

Summary: Model Comparison

Feature	OLS	FE	Standard RE	REWB
Within effect	Blended	✓	Blended	✓
Between effect	Blended	✗	Blended	✓
Time-invariant vars	✓*	✗	✓	✓
Random slopes	✗	Partial	✓	✓
Three+ levels	✗	✗	✓	✓
Level-2 variance	✗	✗	✓	✓
Clustering accounted for	✗	✓	✓	✓

*OLS estimate of z is biased if level-2 omitted variables exist

Key Takeaways

Within ≠ between. Always consider whether these effects should be modelled separately.
REWB is the most general model — it encompasses FE, standard RE, and OLS as special cases.
FE discards valuable information — between effects and higher-level structure — without providing anything REWB cannot.
Omitting random slopes produces anti-conservative SEs — this affects FE and RE equally.
Violating the Normality assumption is usually tolerable — biases in fixed-part estimates are small.
The Hausman test is a test of effect equality, not a model selector.

References

Bell, A., Fairbrother, M., & Jones, K. (2019). Fixed and random effects models: making an informed choice. Quality & Quantity, 53, 1051–1074.

Key works cited in this paper:

Mundlak, Y. (1978). Pooling of time-series and cross-section data. Econometrica, 46(1), 69–85.
Hausman, J.A. (1978). Specification tests in econometrics. Econometrica, 46(6), 1251–1271.
Raudenbush, S.W. & Bryk, A. (2002). Hierarchical Linear Models. Sage.
Allison, P.D. (2014). Problems with the hybrid method. Statistical Horizons.
Barr, D.J. et al. (2013). Random effects structure for confirmatory hypothesis testing. Journal of Memory and Language, 68(3), 255–278.
Reinhart, C.M. & Rogoff, K.S. (2010). Growth in a time of debt. American Economic Review, 100(2), 573–578.

REWB in Practice: Applied Citations

The REWB framework has been adopted across many disciplines:

Political Science

Bell, A. & Jones, K. (2015). Explaining fixed effects: random effects modelling of time-series cross-sectional and panel data. Political Science Research and Methods, 3(1), 133–153. (foundational simulation study)
Jordan, S. & Philips, A.Q. (2023). Improving the interpretation of random effects regression results. Political Science Research and Methods. (interpretive guidance for REWB users)
Stötzer, L. et al. (2025). Does political polarisation undermine democratic accountability? Evidence from 28 European democracies. West European Politics. (REWB with random slopes for TSCS)
Coma, F. & Lago, I. (2025). School subject choices in adolescence affect political party support. West European Politics. (REWB as difference-in-differences analogue)

Sociology & Psychology

Rapp, C. & Stadelmann-Steffen, I. (2020). Policy area satisfaction, perceptions of responsibility, and political trust. Journal of Elections, Public Opinion and Parties. (novel within-respondent REWB application)
Dorrestein et al. (2025). Problematic social media use in individuals with ADHD. (REWB for longitudinal psychopathology outcomes)

Epidemiology & Public Health

Bijnsdorp, F. et al. (2025). Effect of fast-food environments on children’s eating behaviour: Generation R Study. Pediatric Obesity. (REWB for repeated-measures diet data)
Gericke et al. (2022). Delta and Omicron: protective measures and SARS-CoV-2 in day care centres. BMC Infectious Diseases. (REWB panel models for binomial and count outcomes)

Energy Economics

Hirth, L. et al. (2024). Cross-border cannibalization: Spillover effects of wind and solar energy on European electricity markets. arXiv:2405.17166. (REWB to recover between-zone variation lost under FE)