Testing for random effects if fixed effects regression is not possible I have a panel of individuals indexed by $i$, observed at several times $t$. I want to estimate the effect of an individual specific variable, say gender, on an outcome $y_{it}$, either via OLS or random effects regression
$$y_{it}=\alpha+\beta \text{Female}_i+\varepsilon_{it}, \text{or} \\
y_{it}=\alpha_i+\beta \text{Female}_i+\varepsilon_{it}.$$
Clearly, I cannot use a fixed effects regression, because that would absorb the gender effect and I could not estimate $\beta$. Now, how would I test if the random effects assumptions are fulfilled?
Normally, a Hausman test would compare $\beta_{FE}$ and $\beta_{RE}$, but as mentioned, I cannot estimate $\beta_{FE}$.
The Stata manual has implemented a general version of the Hausman test, which states "hausman is a general implementation of Hausman’s (1978) specification test, which compares an estimator $\hat{\theta}_1$ that is known to be consistent with an estimator $\hat{\theta}_2$ that is efficient under the assumption being tested."
So, in my case, since FE is not possible, could I use a simple OLS regression as the consistent estimator, which I can compare to the efficient RE estimator? (This, of course, is assuming OLS is consistent, in the sense there are no important omitted variables correlated with by explanatory variables.)

I am not aware of any test that can help determine if OLS or the GLS random-effects (RE) model is appropriate for your setting.
Another approach is to regress the estimated FE on the female dummy. But that assumes that $\alpha_i$ is uncorrelated with gender to get something unbiased, and you need many observations of the same person ($T \rightarrow \infty$) for consistency. That is equivalent to the assumptions you need for RE to work.
I think what to do next depends on whether you can convince anyone that the RE assumption holds in your setting. I think that is unlikely for most settings with gender since you have nothing else in the model.
