Parameter-Specific Bias Diagnostics in Random-Effects Panel Data Models

The Hausman specification test assesses the random-effects specification by comparing the random-effects estimator with a fixed-effects alternative. This note shows how a recently proposed bias diagnostic for linear mixed models can complement that test in random-effects panel-data applications. The diagnostic delivers parameter-specific internal estimates of finite-sample bias, together with permutation-based $p$-values, from a single fitted random-effects model. We illustrate its use in a gasoline-demand panel and in a value-added model for teacher evaluation using publicly available \textsf{R} packages, and we discuss how the resulting coefficient-specific bias summaries can be incorporated into routine practice.

💡 Research Summary

This paper revisits the classic Hausman specification test for random‑effects (RE) panel models and shows how a recently introduced bias diagnostic for linear mixed models can be used alongside it to assess finite‑sample bias of individual coefficients. The Hausman test compares the RE estimator with a fixed‑effects (FE) alternative; under the null both are consistent and differ only by sampling variation, while under the alternative the RE estimator is inconsistent. The test is global, asymptotic, and focuses on the orthogonality condition between the unobserved individual effects and the regressors.

Karl and Zimmerman (2021) derived an internal bias diagnostic that works when the random‑effects design matrix (Z) is stochastic and possibly dependent on the random effects (\eta). For any linear combination (k’\beta) (a single coefficient or a contrast) they define a weighting vector (\hat\nu_k = k’(X’\hat V^{-1}X)^{-1}X’\hat V^{-1}Z). The finite‑sample bias of the RE estimator for that combination is (E