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?

Sławomir Jarek · Accepted Answer · 2020-03-15T19:09:35.393

0

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.

edited Mar 15 '20 at 19:09

answered Mar 14 '20 at 20:35

Sławomir Jarek

16
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Thank you Slawomir. Assuming no ‘cross-group’ relationship, the within-group correlation should only manifest when $g = g’$. Is this a correct statement? – Thomas Bilach Mar 14 '20 at 21:31
... but the correlation of what? Features or random residues in linear model? – Sławomir Jarek Mar 14 '20 at 22:44
The within-group correlation of the residuals. Should it be written: if $g = g’$? – Thomas Bilach Mar 15 '20 at 01:13
Put differently, the expectation of the residuals is $\sigma_{(ij)}g$, if $g = g’$? – Thomas Bilach Mar 15 '20 at 01:18
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.). – Sławomir Jarek Mar 15 '20 at 16:09
So slide 10 is not the correct formulation? I appreciate the insight! – Thomas Bilach Mar 15 '20 at 17:09
I think so. I think it's a minor editorial mistake. – Sławomir Jarek Mar 15 '20 at 18:52
Thank you. I thought something was awry at first glance. – Thomas Bilach Mar 15 '20 at 19:09
Let us continue this discussion in chat. – Thomas Bilach Mar 15 '20 at 19:25

Within-Cluster Correlation

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