Common mistake 1 with Panel regression

Using random effects because it gives more significant results. Let the test choose.

Common mistake 2 with Panel regression

Forgetting to cluster standard errors. Unclustered panel standard errors are often several times too small.

Common mistake 3 with Panel regression

Claiming fixed effects solves endogeneity. It removes only the TIME-INVARIANT part.

Common mistake 4 with Panel regression

Including a lagged dependent variable with fixed effects - that is Nickell bias, and you need dynamic panel GMM.

EconStatsCraft › Panel regression

Panel regression

Pooled OLS, fixed effects (one-way or two-way), random effects and first differences side by side, with the Hausman and Mundlak tests to choose between FE and RE.

What it needs

Entity identifier - Firm, country, person…
Time identifier - Year, quarter, wave…
Dependent variable
Independent variables

The intuition

If you observe the same firm for ten years, you can compare that firm with ITSELF instead of with other firms. Everything permanently true of it - its culture, its location, its founder - is differenced away, whether or not you ever measured it. That is the single most powerful idea in applied microeconometrics, and its limit is equally clear: anything that CHANGES over time is still a threat.

When to use it

You observe multiple entities across multiple periods and worry about unobserved, time-invariant differences between them.

Where the estimator comes from

Start from y_{it}=\alpha_i+x_{it}'\beta+u_{it}. Average over t within each unit: \bar{y}_i=\alpha_i+\bar{x}_i'\beta+\bar{u}_i. Subtract: (y_{it}-\bar{y}_i)=(x_{it}-\bar{x}_i)'\beta+(u_{it}-\bar{u}_i). The \alpha_i vanish exactly, so OLS on the demeaned data is consistent whatever the correlation between \alpha_i and x. Random effects instead quasi-demeans by \theta=1-\sqrt{\sigma_e^2/(T\sigma_u^2+\sigma_e^2)}, an efficient GLS weighting that is only valid if \operatorname{Cov}(\alpha_i,x_{it})=0. Hausman compares the two: if RE is consistent both estimate the same thing, so a large gap indicts RE.

The model

y_{it}=\alpha_i+\mathbf{x}_{it}^{\prime}\boldsymbol{\beta}+u_{it}\ \ \xrightarrow{\text{within}}\ \ (y_{it}-\bar{y}_i)=(\mathbf{x}_{it}-\bar{\mathbf{x}}_i)^{\prime}\boldsymbol{\beta}+(u_{it}-\bar{u}_i)

What it assumes

Fixed effects: the unobserved effect is constant over time and may correlate with x.
Random effects: the unobserved effect is UNCORRELATED with x. Strong, and usually wrong.
Both: no time-varying omitted variable. Fixed effects does not solve that.
Errors are usually correlated within a unit - cluster the standard errors by entity.

Statistical properties

FE is consistent whether or not αᵢ correlates with x; RE is consistent only if it does not, but is more efficient when valid.
FE cannot estimate the effect of anything time-invariant - those coefficients are absorbed, not zero.
θ = 0 makes RE identical to pooled OLS; θ = 1 makes it identical to FE.
Errors are almost always correlated within a unit, so cluster by the unit. Unclustered panel standard errors are commonly several times too small.
Cluster-robust inference is asymptotic in the NUMBER OF CLUSTERS, so it needs roughly 40+ clusters; the reference distribution is t(G−1).
FE with a lagged dependent variable is inconsistent for small T - Nickell bias, of order 1/T - and needs Arellano–Bond.

How to read the output

Compare pooled OLS and FE. A large gap means unobserved heterogeneity was biasing the pooled estimate.
Hausman/Mundlak rejects → use fixed effects. Fails to reject → random effects is more efficient.
Fixed effects cannot estimate anything that never changes within a unit - gender, country of birth. Those get absorbed.
Two-way fixed effects also removes anything common to all units in a period, like a national recession.
R² within is the relevant fit measure for FE.

A worked reading

Pooled OLS puts the capital elasticity at 0.95; one-way FE at 0.42; two-way FE at 0.34, close to the true 0.32. The pooled estimate was picking up permanently more productive firms using more capital, and the remaining one-way FE bias was a common time trend that only two-way FE removes.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Pooled and FE differ substantially	Unobserved heterogeneity correlated with x	Use FE and say why
Hausman rejects	RE inconsistent	Fixed effects
Hausman statistic is negative	RE is not efficient, so the test is invalid	Use the Mundlak version, which is robust
Key regressor barely varies within units	FE has almost nothing to identify from	Report the wide interval; consider a between or RE estimator
Fewer than ~40 clusters	Cluster SEs over-reject	Wild cluster bootstrap
Lagged y among the regressors	Nickell bias	Arellano–Bond / system GMM

Common mistakes

Using random effects because it gives more significant results. Let the test choose.
Forgetting to cluster standard errors. Unclustered panel standard errors are often several times too small.
Claiming fixed effects solves endogeneity. It removes only the TIME-INVARIANT part.
Including a lagged dependent variable with fixed effects - that is Nickell bias, and you need dynamic panel GMM.

Where to read this properly

Wooldridge (2010), Econometric Analysis of Cross Section and Panel Data, 2nd ed., MIT Press, chapters 10-11
Mundlak (1978)
Hausman (1978)
Nickell (1981)
Cameron, Gelbach & Miller (2008) - the wild cluster bootstrap

Related Panel data

Panel cointegration tests · Heterogeneous panel: Mean Group & PMG · Gravity model of trade · Dynamic panel GMM · Panel diagnostics · Panel unit-root tests

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Panel regression

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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