Questions: Between and Random Effects Estimators for Panel Data

Question 1

You are studying the effect of job training programs on wages using panel data. You suspect that workers who self-select into training are systematically more motivated — and motivation is unobserved but positively correlated with wages. Which estimator should you use?

• A. The between estimator, since it uses cross-sectional differences between trained and untrained workers
• B. The random effects estimator, which is more efficient than fixed effects
• C. The within (fixed effects) estimator, which removes time-invariant unobserved heterogeneity including motivation
• D. OLS on pooled data, since motivation averages out across the full sample

Answer: (C)

The motivation-wage correlation means the orthogonality assumption (αᵢ ⊥ Xᵢₜ) fails: the unobserved unit-specific component (motivation) is correlated with the regressor (training). Both the between estimator and random effects would be inconsistent in this case — they rely on cross-unit variation in training, which is confounded with cross-unit variation in motivation. The within estimator removes all time-invariant unit-level effects by demeaning, including the unobserved motivation component, isolating only within-worker variation in training over time. This is the key strength of fixed effects.

Question 2

The between estimator for panel data is computed by:

• A. Regressing each observation's deviation from its unit time-mean on the within-unit deviation of regressors
• B. Running OLS on the time-averaged observations for each unit (group means)
• C. Running OLS on first differences between consecutive time periods
• D. Applying GLS weights that blend within- and between-unit variation

Answer: (B)

The between estimator collapses each unit's panel observations into a single row of time-averaged values (ȳᵢ, X̄ᵢ) and then runs OLS on those averages. The result is estimated entirely from cross-sectional variation — how units with higher average X differ from units with lower average X. This is the 'opposite' of fixed effects: where fixed effects removes between-unit variation by demeaning, the between estimator uses only between-unit variation and discards the within-unit time-series information entirely.

Question 3

The random effects estimator is more efficient than fixed effects when the unit-specific effect is uncorrelated with all regressors.

• T. True
• F. False

Answer: True

When αᵢ ⊥ Xᵢₜ holds, both fixed effects and random effects are consistent, but random effects uses more information. Fixed effects discards all between-unit variation (by demeaning), throwing away valid identifying variation. Random effects uses GLS to optimally combine both within- and between-unit variation, producing more precise estimates. The efficiency gain is real and can be substantial when between-unit variation is large — but it disappears entirely if the orthogonality assumption fails.

Question 4

If random effects and fixed effects produce very similar coefficient estimates, this is evidence that the orthogonality assumption (αᵢ ⊥ Xᵢₜ) has likely failed.

• T. True
• F. False

Answer: False

This is the reverse of the correct logic. The Hausman test detects failure of the orthogonality assumption by asking whether random effects and fixed effects estimates are *statistically different*. When the assumption holds, both estimators are consistent and should converge to similar values — similar estimates are evidence the assumption is satisfied. A large, statistically significant difference between the two estimates is the warning sign that random effects may be inconsistent and fixed effects should be trusted.

Question 5

Explain the trade-off between random effects and fixed effects estimators. Under what condition does random effects fail, and why does fixed effects remain valid in that case?

Model answer: Random effects assumes the unit-specific effect αᵢ is uncorrelated with all regressors (αᵢ ⊥ Xᵢₜ). When this holds, random effects uses GLS to blend within- and between-unit variation, yielding more efficient estimates than fixed effects. When the assumption fails — when unobserved unit characteristics are correlated with the regressors (e.g., unobserved firm quality correlated with investment) — random effects is inconsistent, meaning its estimates are biased even in large samples. Fixed effects remains valid because it eliminates αᵢ entirely by demeaning: the within transformation ÿᵢₜ = yᵢₜ − ȳᵢ removes all time-invariant unit-level variation, including the unobserved correlated component. The cost is that you lose all information from cross-unit differences and cannot estimate coefficients on time-invariant regressors.

The trade-off is efficiency vs. robustness. Random effects wins on efficiency (uses more variation) but requires a strong assumption. Fixed effects is more robust (doesn't require the orthogonality assumption) but wastes potentially valid identifying variation. The Hausman test helps choose between them empirically.