The Hausman test can be used to choose between two estimators where one is less efficient but consistent under both alternatives whereas the other is more efficient but only consistent under the null hypothesis.

The Hausman test can be used to choose between two estimators where one is less efficient but consistent under both alternatives whereas the other is more efficient but only consistent under the null hypothesis. The test statistic is $$H = (\beta_{a}-\beta_{b})'[Var(\beta_{a})-Var(\beta_{b})]^{-1}(\beta_{a}-\beta_{b})$$ where $a$ is the consistent estimator and $b$ is the efficient estimator. Typical examples for such comparisons are IV v.s. OLS and fixed effects v.s. random effects in panel data.

For more information on heteroscedasticity robust, cluster robust or bootstrapped versions of the test see

  • Cameron and Trivedi (2010) "Microeconometrics Using Stata", Stata Press
  • Wooldridge (2010) "Econometric Analysis of Cross Section and Panel Data", MIT Press