Within-Cluster Correlation

Question

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?

Answer 1

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.