# Panel Data- FE vs RE

I am trying to understand the different Panel Data models and I am getting confused by the different terms that people use, i.e., Random effect models and Random effects estimators and Fixed effect models and Fixed effect estimators- are these 4 all different?

Is it the case that you can have a fixed effect estimator in a random effects model and if you do that your estimate is not consistent?

For context, I am running a panel data analysis in STATA and after conducting a Hausman Test my p-value is more that 0.05 indicating I should use a random effects model. Does this mean I simply use the ,re output as my result or does it mean there's something in my variables and controls that needs to be changed or considered?

I am running a regression of total inflow migration on gov spending on healthcare per capita. My other x variables are population density, share of elderly, total tax revenue, GDP per capita and gini.

I understand that I am not to include variables that do not vary over time within a country- or is that only for fixed effects? So I'm not sure if my x variables are 'correct'

I'm sorry these are quite a few questions but any help at all would be GREATLY appreciated. The more that I try to read about panel data models, the more I get myself confused so if someone could explain for my specific case that would be amazing. Thank you!!

FE estimator: A particular estimator, also known as 'within-group' estimator, usually for linear panel data models with strictly exogenous regressors; can be computed by Stata's xtreg, fe.
RE estimator: A particular FGLS estimator for the linear error component model (the error term = individual effects ($$u_i$$) + idiosyncratic error ($$e_{it}$$), where $$u_i$$ and $$e_{it}$$ are mutually uncorrelated, $$e_{it}$$ are homoskedastic and serially uncorrelated); can be computed by Stata's xtreg, re.