ECMT3120 Chap.8 Fixed Effects, Random Effects and Model Choice
Fixed Effects, Random Effects and Model Choice
Fixed effects and random effects use panel structure in different ways. Fixed effects removes unit means and permits stable unit heterogeneity to correlate with regressors, while random effects uses a quasi-demeaning transformation and requires orthogonality between the unit effect and the regressors.
This chapter compares their identifying variation, derives the random effects transformation parameter, explains efficiency claims, and treats the Hausman comparison as a diagnostic with maintained assumptions rather than a mechanical estimator-selection button.
What this chapter covers
- 01
Within transformation for fixed effects
- 02
Identification from deviations around unit means
- 03
Loss of time-invariant regressors under fixed effects
- 04
Composite error structure in random effects
- 05
Quasi-demeaning and the transformation parameter
- 06
Random effects orthogonality assumption
- 07
Efficiency and feasible generalized least squares
- 08
Hausman comparison and correlated random effects
Worked example · free
Compute a random effects transformation
- +2Use lambda equals one minus the square root of sigma u squared divided by sigma u squared plus T sigma a squared.
- +1Substitute the variance components to obtain one minus the square root of 4 over 7.
- +1Calculate lambda as approximately 0.244.
- +1Explain that each variable is transformed as its observation minus lambda times its unit mean.
- +1When the unit-effect variance approaches zero, lambda approaches zero and the transformation approaches pooled OLS.
- +2When the unit effect dominates or the panel becomes long, lambda approaches one and the transformation approaches fixed effects demeaning.
Key terms
- Fixed effects estimator
- An estimator based on deviations from unit means that allows regressors to correlate with stable unit effects.
- Within transformation
- Subtracting each unit's time average from every observation for that unit.
- Random effects estimator
- A feasible generalized least squares estimator using both within and between variation under an orthogonality assumption.
- Quasi-demeaning
- Subtracting a fraction of the unit mean determined by estimated variance components.
- Variance component
- A portion of the composite error variance attributed to the stable unit effect or idiosyncratic disturbance.
- Hausman comparison
- A comparison of coefficient vectors based on the idea that both estimators are consistent under the stronger random effects assumptions.
Fixed Effects, Random Effects and Model Choice FAQ
Why can fixed effects include a time effect but not a unit-constant regressor?
A common time indicator changes across periods and survives unit demeaning. A unit-constant regressor equals its own unit mean in every period, so its within-transformed value is zero and its separate coefficient cannot be recovered.
Is random effects always more efficient than fixed effects?
Only under its stronger orthogonality and covariance assumptions. If the unit effect correlates with regressors, random effects generally becomes inconsistent, so a smaller reported standard error is not a meaningful advantage.
Does a Hausman rejection prove fixed effects is the correct causal model?
No. Rejection indicates a systematic coefficient difference under the test's maintained assumptions. It can reflect correlation with unit effects, other misspecification or unreliable covariance estimates. Substantive design reasoning still determines the credible specification.
Exam move
Build a comparison table with rows for transformation, source of variation, treatment of stable heterogeneity, time-invariant regressors and key orthogonality assumptions. Derive fixed effects by subtracting unit means and random effects by subtracting lambda times those means. Practise the limiting cases of lambda so the formula has economic meaning.
When choosing an estimator, state consistency conditions before discussing efficiency. Interpret the Hausman comparison through its null and maintained assumptions instead of quoting a p value alone. As an extension, write a correlated random effects specification that adds unit means of time-varying regressors, making the disputed correlation visible and testable.
For each empirical story, name a stable unit trait and decide whether it plausibly correlates with the changing regressor. Use that answer to motivate the transformation before looking at standard errors. Compute lambda from several variance combinations and check the endpoints: negligible unit heterogeneity should resemble pooled levels, while dominant unit heterogeneity or a long panel should approach full demeaning.
Transform the intercept as well as the observed variables so the random-effects equation remains internally consistent. In fixed-effects exercises, list every time-invariant regressor that disappears and state which substantive question can no longer be answered.
Treat a Hausman rejection as evidence of systematic coefficient disagreement under maintained covariance conditions, not proof that all remaining fixed-effects assumptions hold. Finally, compare within, between, quasi-demeaned and adjacent-change variation in plain language so the reported coefficient is tied to the observations that identify it.
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