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Let's say that we have this simple model:

\begin{equation} \label{eq:gls_reg} y_{ij} = x_{ij}\cdot\beta +{u_{i}} + \varepsilon_{ij} , \end{equation} \begin{equation*} u_i\sim N\left( {0,\sigma_{u} ^2 } \right) , \end{equation*} \begin{equation*} \varepsilon_{ij} \sim {\rm N}\left( {0,{\sigma_{\varepsilon} ^2}} \right) , \end{equation*} thus $u_i$'s are idd and $\varepsilon_{ij}$'s are iid.

Are $u_i$'s independent of $\varepsilon_{ij}$'s?

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It is assumed that $u_i$ and $\varepsilon_{ij}$ are independent of each other and in fact they should not just be mutually independent but also independent of $x_{ij}$ (see Verbeek (2008). A guide to modern econometrics. 4th ed. pp 381).

Also $u_i + \varepsilon_{ij}$ is treated as an error term consisting of two components where the first part is individual time-invariant component and second part is a remainder component that is not correlated over time.

Of course, these are assumptions of the model, whether they actually hold in the data is something that should be tested or justified with some theoretical reasoning.

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