Pooled GLS and POLS

Hello everyone,

This semester I'm studying, OLS, GLS, RE, FE, MLE, and a lot of beautiful concepts :D. But I get a little bit concerning the different assumptions of each model and their difference.

Here is what I need to answer and I will say what I've understood/found so far thanks my friends Wooldridge and Trivedi:

Consider the linear panel data model in matrix notation:
yi =Xiβ+ui
where yi and ui are T ×1 vectors and Xi is a T ×K matrix.

a) Describe and motivate the pooled GLS estimator. What assumptions are needed for consistency and efficiency of pooled GLS? How do they differ from those for POLS?

OLS required exogeneity (no correlation between the regressors and the error term) and full rank, so no multicollinearity among the regressors. With those two, you can say that OLS is consistent. You had homoskedasticity, variance of the error term doesn't vary for all values of independent variables, and you get all the assumptions.

The problem is I've found the same assumptions for GLS. I guess I don't get why we use GLS, instead it is supposed to be "more strict".

I thank you in advance for your help ;)