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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!!

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RE model: panel data model with some restrictions on the individual effects (such as being uncorrelated with the regressors for linear panel data models).

FE model: panel data model without such restrictions (so the individual effects can be correlated with the regressors).

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.

FE estimator is consistent for both FE and RE models. RE estimator is consistent for RE models but inconsistent for FE models in general.

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