which method random effects, fixed effects, or pooled OLS estimation should I use? 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? 
 A: The random effects estimator assumes that the fixed effect predictors are uncorrelated with the random intercept. This is potentially problematic for predictors that are measured at the same level as the intercept (e.g., sex or gender when persons are given a random intercept). It is not always the case that person-level variables are correlated with the random intercept, but in order to use the RE model, you must make the assumption that they are not correlated. Thus, as long as you are making that assumption clear, you could use a RE model. Of course, you could use the FE model and not have to make that assumption. 
But, as you say, the FE model will not allow you to estimate the wage gap, so you must use either pooled OLS, which we know is problematic given the repeated measure data, or a RE model, which is appropriate but requires you to make the assumption clear.  
One thing to keep in mind is that occasion-specific predictors (e.g., experience) have both within- and between-person variation, and you can disentangle this variation in one of two ways:


*

*Calculate the mean of the occasion-specific variable(s) for each person and include that mean as a predictor in your random effects model along with the original variable (e.g., exp and the person mean of exp).

*Create a person mean-centered predictor by subtracting a person's value on an occasion-specific variable from their value on a given occasion for that variable. Include the person-centered version of the variable as the occasion-specific predictor along with the person mean.


The coefficient for the person mean in #1 is actually a test of whether the within and between effects of a occasion-varying variable are different from one another. If the coefficient is significant, then the effects are different and you may have endogeneity (correlation between that person-level variable and the random intercept). If that coefficient is not significant, then the within and between effects are equivalent and the RE and FE models are equivalent. 
