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I figured this would be more appropriate on this forum. I came across the following slides after a Google search of cluster-robust uncertainty estimators. On slide 10 the author attempts to demonstrate how the errors of students nested within the same classroom are correlated. That is, $\textrm{E}\left[u_{ig} u_{jg'}\right] = \sigma_{(ij)g}$ if $g \neq g'$.

Shouldn't this be: if $g = g'$? I assume $g \neq g'$ represents two students ($i$ and $j$) in different classrooms (groups). If so, wouldn't this be demonstrative of dependence across clusters?

Sorry if this question is a bit in the weeds. Can anyone provide further clarity?

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Note that $u_i$ is a random residue. In the linear regression model, we assume the independence of the random residue (error term). We have two slides above for case i.i.d.: $\epsilon_i \sim (0,\sigma^2)$, thus $\forall_{i\neq j}E[\epsilon_i\epsilon_j|X]=0$ and $E[\epsilon_i\epsilon_i|X]=\sigma^2$. Later (slide 10) we assume i.n.i.d. and $\forall_{g\neq g'}E[\epsilon_{ig}\epsilon_{jg'}]=0$.

Slide 12 shows the example matrix $\Omega=diag(\Sigma_g)$ that dispels doubts.

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  • $\begingroup$ Thank you Slawomir. Assuming no ‘cross-group’ relationship, the within-group correlation should only manifest when $g = g’$. Is this a correct statement? $\endgroup$ – Tom Mar 14 at 21:31
  • $\begingroup$ ... but the correlation of what? Features or random residues in linear model? $\endgroup$ – Sławomir Jarek Mar 14 at 22:44
  • $\begingroup$ The within-group correlation of the residuals. Should it be written: if $g = g’$? $\endgroup$ – Tom Mar 15 at 1:13
  • $\begingroup$ Put differently, the expectation of the residuals is $\sigma_{(ij)}g$, if $g = g’$? $\endgroup$ – Tom Mar 15 at 1:18
  • $\begingroup$ You are right Tom, the order of the lines has been changed there. In this case, we have residuals ​​that are independent, non-identically distributed (i.n.i.d.). $\endgroup$ – Sławomir Jarek Mar 15 at 16:09

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