Consider a model at the employee level, $$y_{i, e}=\beta_{0}+\beta_{1} x_{i, e, 1}+\beta_{2} x_{i, e, 2}+\cdots+\beta_{k} x_{i, e, k}+f_{i}+v_{i, e}$$ where the unobserved variable $f_{i}$ is a "firm effect" to each employee at a given firm $i .$ The error term $v_{i, e}$ is specific to employee $e$ at firm $i .$ The composite error is $u_{i, e}=f_{i}+v_{i, e},$ such as in equation (8.28)
i. Assume that $\operatorname{Var}\left(f_{i}\right)=\sigma_{f}^{2}, \operatorname{Var}\left(v_{i, e}\right)=\sigma_{v}^{2},$ and $f_{i}$ and $v_{i, e}$ are uncorrelated. Show that $\operatorname{Var}\left(u_{i, e}\right)=\sigma_{f}^{2}+\sigma_{v}^{2} ;$ call this $\sigma^{2}$
ii. Now suppose that for $e \neq g, v_{i, e}$ and $v_{i, g}$ are uncorrelated. Show that $\operatorname{Cov}\left(u_{i, e}, u_{i, g}\right)=\sigma_{f}^{2}$
iii. Let $\bar{u}_{i}=m_{i}^{-1} \sum_{i=1}^{m i} u_{i, e}$ be the average of the composite errors within a firm. Show that
$$\operatorname{Var}\left(\bar{u}_{i}\right)=\sigma_{f}^{2}+\sigma_{v}^{2} / m_{i}$$.
iv. Discuss the relevance of part (iii) for WLS estimation using data averaged at the firm level, where the weight used for observation $i$ is the usual firm size.