Within-Cluster Correlation

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?

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.
• Thank you Slawomir. Assuming no ‘cross-group’ relationship, the within-group correlation should only manifest when $g = g’$. Is this a correct statement? – Tom Mar 14 at 21:31
• The within-group correlation of the residuals. Should it be written: if $g = g’$? – Tom Mar 15 at 1:13
• Put differently, the expectation of the residuals is $\sigma_{(ij)}g$, if $g = g’$? – Tom Mar 15 at 1:18