I understand the assumption for using a random effects model is that the unobserved group-level effects are uncorrelated with the explanatory variables. I wonder if these effects refer to such time-invariant heterogeneities as gender and country of origin, the omitting of which can result in biased coefficients.

I also wonder if some time-invariant explanatory variables have strong effects on the outcome variable, whether using a fixed effects model will produce very imprecise coefficients.

Which scenario is worse, biased or imprecise? In both cases, the coefficients do not accurately reflect the effect of the predictor variable on the outcome variable. Is it in the first case, the direction of the coefficient is correct, but the magnitude is wrong? And in the second case, is it just a wrong coefficient?

Thank you.