How does the Hausman test choose between fixed and random effects in panel data?

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AcadiFi · Accepted Answer

Fixed effects (FE) and random effects (RE) both estimate panel data models, but they rest on different assumptions. FE allows each entity to have its own intercept and is consistent even if unobserved entity characteristics correlate with the regressors. RE treats those unobserved characteristics as random draws uncorrelated with regressors, which is more efficient but biased if the assumption fails.

The Hausman test formalizes the trade-off. Under the null that entity effects are uncorrelated with regressors, both estimators are consistent but RE is efficient. Under the alternative, only FE is consistent. The statistic compares FE and RE coefficient vectors: H = (b_FE - b_RE)' [Var(b_FE) - Var(b_RE)]^(-1) (b_FE - b_RE). It follows a chi-squared distribution with degrees of freedom equal to the number of time-varying regressors.

A large H value rejects RE's exogeneity assumption and recommends FE. A small H suggests RE is acceptable and more efficient.

Imagine econometrician Ines Hollister modeling Montclear Industries and 39 peers' capital expenditure on cash flow, leverage, and Tobin's Q. Hausman statistic is 18.7 with 3 degrees of freedom, p-value 0.0003. She rejects RE and uses FE.

Caveats: the test assumes proper specification otherwise. If important time-invariant variables are excluded from FE (since FE eliminates them), you lose their effects entirely. The Mundlak approach offers a middle path by including entity means as additional regressors, allowing you to test and potentially avoid the dichotomy.

On the CFA exam, recognize the null of both consistency and the H-statistic interpretation.

How does the Hausman test choose between fixed and random effects in panel data?

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