I am working on a project with a time invariant independent variable (childhood nutrition at age 5). I have so far been working with a pooled OLS model with clustered robust standard errors at the panel (Household) level as my dataset is panel in nature-with clustered S.Es to correct for serial autocorrelation of the errors within a cluster (household) across time). I have 2000 clusters with 5 observations per cluster.

I cannot use a fixed effects due to the time invariant nature of my main independent variable. I was hoping to compare my results with pooled OLS with clustered robust standard errors with a random effects model. I was having some trouble understanding the relative benefits of doing so as from reading, my understanding is both models aim to correct for serial correlation and heteroscedasticity in errors across time. I have 2 specific questions:

1) What are the benefits of Random Effects over Pooled OLS with clustered robust standard errors ? Do they correct for different phenomena? (Any resources i.e. links you might be able to provide would be useful here
2) In STATA, what does the robust command do with random effects? Does not using robust with random effects correct for serial correlation anyway?