I'm stuck on a question since I'm confused about some of the assumptions that need to be satisfied to use these methods.
The assumptions that need to be satisfied given in our lecture notes is that for the pooled OLS and RE estimator we need that the individual effects and regressors cannot be correlated, and for FE this assumption is relaxed but in our book it says that time-constant variables cannot be included for the FE method (but can be included in RE or pooled OLS).
The question uses the following model:
a) Is random effects or fixed effects estimator appropriate in this case. Explain your answer using underlying assumptions.
b) Let the main interest be in estimating the wage gap between men and women so the coefficient β6. What estimation method can you use (fixed effects, random effects or pooled OLS). Justify your answer using underlying assumptions.
My issue is that in part (a) I concluded that the FE estimator is appropriate since there is likely a correlation between the individual effects and the regressors making RE unsuitable. But when I moved to part (b), part (a) implies RE and pooled OLS are unsuitable, but I also concluded that FE was unsuitable since "female" is also a time-constant variable, and that cannot be included in the FE model. Which then means that in part (a) I should have also concluded that the FE estimator is unsuitable, meaning all the options are unsuitable for both questions!
What's going wrong in my thinking?