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2

Consider the following specification: $$ Y_{i,g} = X_{i,g}\beta + u_{i,g} $$ Where the residuals have different mean across groups and have within group correlation: $$ \begin{align*} &\mathbb{E}(u_{i,g}) = \alpha_g,\\ &cov(u_{i,g} u_{j,g}) = \rho_{i,j},\\ &cov(u_{i,g}, u_{j,g'}) = 0 \text{ for } g \ne g' \end{align*} $$ Taking means gives: $$ \...


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Usually, the conditional mean and the conditional variance of a random variable are independent (of course there are exceptions). While the fixed effects introduce flexibility in the specification of the conditional mean, clustering introduce flexibility in the specification of the conditional variance.


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I guess this follows from the Frish-Waugh Lovell theorem. If you have $K$ dummies and say $n$ observations, so $F$ is $n \times K$, you need to regress every dummy in $F$ on all variables in $X$ separately ($K$ regressions). The residuals make up the matrix $\bar F$ which is again of dimension $n \times K$.


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