Common mistake 1 with Panel diagnostics

Assuming clustering fixes everything. It handles within-entity correlation, not cross-sectional dependence.

Common mistake 2 with Panel diagnostics

Running the modified Wald on very short panels, where the variance estimates are too noisy to trust.

Common mistake 3 with Panel diagnostics

Treating a failed test as fatal. Most point to a standard-error fix, not a broken model.

Common mistake 4 with Panel diagnostics

Running every test and reporting only the ones that pass. These tests overlap, and selecting among them after seeing the results is how a misspecified model acquires a clean bill of health.

EconStatsCraft › Panel diagnostics

Panel diagnostics

The four tests that decide whether your panel specification is defensible: are there entity effects at all, are the errors serially correlated, does error variance differ across entities, and are the entities hit by common shocks?

What it needs

Entity identifier
Time identifier
Dependent variable
Independent variables

The intuition

Before choosing a panel estimator, ask whether the panel structure matters at all. Three separate questions, each with its own test: is there anything entity-specific to worry about, are the entity effects correlated with the regressors, and are the errors behaving. Answer them in that order, because the answer to the first can make the rest moot.

When to use it

After any panel regression, before you report it. Each test maps to a specific fix, and all four are cheap.

Where the estimator comes from

The Breusch-Pagan LM test asks whether \sigma_\alpha^2=0 in u_{it}=\alpha_i+\varepsilon_{it}; the statistic is built from the within-group sums of pooled OLS residuals and is \chi^2_1 under the null of no random effect. The Hausman test compares two estimators that are both consistent under the null and only one under the alternative: H=(\hat{\beta}_{FE}-\hat{\beta}_{RE})'\left[V(\hat{\beta}_{FE})-V(\hat{\beta}_{RE})\right]^{-1}(\hat{\beta}_{FE}-\hat{\beta}_{RE})\sim\chi^2_k. The variance difference is what makes it fragile - it can come out non-positive-definite in finite samples. The Mundlak formulation avoids that by adding group means \bar{x}_i to the random-effects regression and testing their joint significance, which is the same hypothesis with a well-behaved Wald test.

The model

\text{CD}=\sqrt{\frac{2}{N(N-1)}}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\sqrt{T_{ij}}\;\hat{\rho}_{ij}

What it assumes

The regression is otherwise correctly specified - these test the error structure, not the functional form.
Breusch–Pagan LM assumes a balanced panel for its exact distribution; it is approximate otherwise.
Wooldridge needs at least three periods per entity.
The panel is long enough for the tests requested. Serial-correlation tests on three or four periods per entity have almost no power, so a clean result there is uninformative rather than reassuring.

Statistical properties

The Mundlak test and the Hausman test address the same null. Mundlak is preferable in practice: it never produces a negative statistic, extends naturally to robust and clustered variances, and shows WHICH regressors are correlated with the effect rather than only that some are.
A negative Hausman statistic is not evidence for the null. It means the variance difference is not positive definite in this sample, and the test is simply uninformative.
Rejecting Hausman does not mandate fixed effects. It says random effects is inconsistent under its own assumptions; if the coefficient of interest is on a time-invariant variable, fixed effects cannot estimate it at all and correlated random effects is the way out.
The classical Hausman test assumes random effects is EFFICIENT under the null, which fails the moment errors are clustered or heteroskedastic. With robust standard errors the classical form is invalid and the Mundlak version should be used.
Pesaran's CD test detects cross-sectional dependence - common shocks hitting all entities. It is very common in country and industry panels and invalidates ordinary clustered standard errors, which assume independence across groups.
Groupwise heteroskedasticity is nearly universal in firm panels and is not a specification error; it is an argument for clustered or robust variances rather than for a different estimator.

How to read the output

Breusch–Pagan LM rejects → entity effects exist, so pooled OLS is wrong.
Wooldridge rejects → serial correlation, so cluster standard errors by entity.
Modified Wald rejects → error variance differs across entities; cluster again.
Pesaran CD rejects → common shocks. Clustering by entity does NOT fix this; use Driscoll–Kraay or add time effects.
The summary at the bottom turns whichever tests rejected into a single instruction.

A worked reading

On a firm panel the LM test rejects decisively, so there is an entity effect. Hausman then rejects at p < 0.001 and the Mundlak test agrees, with the group mean of capital carrying almost all of the significance - which is more useful than Hausman's verdict, because it names the regressor that is correlated with the fixed effect rather than just announcing that one is. Pesaran CD also rejects, so the firms share common shocks: clustering by firm is not sufficient here, since it assumes independence ACROSS firms, and that is exactly what CD has just refuted.

Symptom → diagnosis → what to do

If you see this	It means	Do this
LM test not rejecting	No entity effect worth modelling	Pooled OLS is adequate. Stop here
Hausman statistic negative	Variance difference not positive definite	Uninformative. Use the Mundlak form
Hausman rejects but you need a time-invariant coefficient	Fixed effects cannot estimate it	Use correlated random effects
Pesaran CD rejecting	Common shocks across entities	Clustered standard errors are not enough; use Driscoll-Kraay
Strong groupwise heteroskedasticity	Normal in firm data	Cluster by entity; do not change estimator
Serial correlation in the within residuals	Wooldridge test rejecting	Cluster by entity, which handles both at once

Common mistakes

Assuming clustering fixes everything. It handles within-entity correlation, not cross-sectional dependence.
Running the modified Wald on very short panels, where the variance estimates are too noisy to trust.
Treating a failed test as fatal. Most point to a standard-error fix, not a broken model.
Running every test and reporting only the ones that pass. These tests overlap, and selecting among them after seeing the results is how a misspecified model acquires a clean bill of health.

Where to read this properly

Hausman (1978), "Specification Tests in Econometrics", Econometrica 46(6).
Mundlak (1978), "On the Pooling of Time Series and Cross Section Data", Econometrica 46(1).
Pesaran (2004, 2021), on cross-sectional dependence.
Wooldridge, Econometric Analysis of Cross Section and Panel Data (2010), ch. 10.

Related Panel data

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

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Panel diagnostics

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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