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I am trying to see what model performs best. I am considering the Hausman test to see which model is the most stable out of the three. However, I am not sure what exactly my null hypothesis should be when I test three models, and if I should compare RE against Pooled OLS and then RE against FE.

Do you think that this is the correct way of testing? Also, is there another test or statistic that can be used in my case?

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If the time-invariant variation is relevant to your identification then RE of FE is needed.

You can compare OLS to FE with something like an F test of significance of the extra dummies.

The Hausman test compares a consistent estimator to an efficient one. In this context, fixed effects would be consistent (as it controls for the time-invariant OVB) while random effects is relatively efficient.

However be careful with the Hausman as it won't tell you whether the time invariant variation is orthogonal to your independent variables. That's more of a judgement call.

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