<p>Note that <span class="math-container">$u_i$</span> 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.: <span class="math-container">$\epsilon_i \sim (0,\sigma^2)$</span>, thus <span class="math-container">$\forall_{i\neq j}E[\epsilon_i\epsilon_j|X]=0$</span> and <span class="math-container">$E[\epsilon_i\epsilon_i|X]=\sigma^2$</span>. Later (slide 10) we assume i.n.i.d. and <span class="math-container">$\forall_{g\neq g'}E[\epsilon_{ig}\epsilon_{jg'}]=0$</span>.</p>

<p>Slide 12 shows the example matrix <span class="math-container">$\Omega=diag(\Sigma_g)$</span> that dispels doubts.</p>

CC BY-SA 4.0
2020-03-14T21:31:07.750Z
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
CC BY-SA 4.0
2020-03-14T22:44:35.277Z
... but the correlation of what? Features or random residues in linear model?
Sławomir Jarek
CC BY-SA 4.0
2020-03-15T01:13:30.653Z
The within-group correlation of the residuals. Should it be written: if $g = g’$?
Thomas Bilach
CC BY-SA 4.0
2020-03-15T01:18:12.337Z
Put differently, the expectation of the residuals is $\sigma_{(ij)}g$, *if* $g = g’$?
Thomas Bilach
CC BY-SA 4.0
2020-03-15T16:09:26.303Z
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
CC BY-SA 4.0
2020-03-15T17:09:23.213Z
So slide 10 is not the correct formulation? I appreciate the insight!
Thomas Bilach
CC BY-SA 4.0
2020-03-15T18:52:43.943Z
I think so. I think it's a minor editorial mistake.
Sławomir Jarek
CC BY-SA 4.0
2020-03-15T19:09:19.133Z
Thank you. I thought something was awry at first glance.
Thomas Bilach
CC BY-SA 4.0
2020-03-15T19:25:54.200Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/105585/discussion-between-tom-and-slawomir-jarek).
Thomas Bilach
CC BY-SA 4.0
2020-03-14T20:35:49.903Z
2020-03-15T19:09:35.393Z
2020-03-15T19:09:35.393Z
Sławomir Jarek
<p>I figured this would be more appropriate on this forum. I came across the following <a href="http://econweb.umd.edu/~sarzosa/teach/2/Disc2_Cluster_handout.pdf" rel="nofollow noreferrer">slides</a> after a Google search of cluster-robust uncertainty estimators. On slide 10 the author attempts to demonstrate how the errors of students nested <em>within</em> the same classroom are correlated. That is, <span class="math-container">$\textrm{E}\left[u_{ig} u_{jg'}\right] = \sigma_{(ij)g}$</span> if <span class="math-container">$g \neq g'$</span>.</p>

<p>Shouldn't this be: if <span class="math-container">$g = g'$</span>? I assume <span class="math-container">$g \neq g'$</span> represents two students (<span class="math-container">$i$</span> and <span class="math-container">$j$</span>) in <em>different</em> classrooms (groups). If so, wouldn't this be demonstrative of dependence across clusters?</p>

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

<applied-econometrics>
Within-Cluster Correlation
Thomas Bilach
# Within-Cluster Correlation

(Asked by: username_0 on 2020-03-13)

I figured this would be more appropriate on this forum. I came across the following [slides](http://econweb.umd.edu/~sarzosa/teach/2/Disc2_Cluster_handout.pdf) 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?

**Tags:** applied-econometrics


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## 1 answers

### Answer by username_1 on 2020-03-14. Score: 0 | Accepted answer

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.

> **Comments:**
> **username_0**: Thank you Slawomir. Assuming no ‘cross-group’ relationship, the within-group correlation should only manifest when $g = g’$. Is this a correct statement?  
> **username_1**: ... but the correlation of what? Features or random residues in linear model?  
> **username_0**: The within-group correlation of the residuals. Should it be written: if $g = g’$?  
> **username_0**: Put differently, the expectation of the residuals is $\sigma_{(ij)}g$, *if* $g = g’$?  
> **username_1**: 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.).  
> **username_0**: So slide 10 is not the correct formulation? I appreciate the insight!  
> **username_1**: I think so. I think it's a minor editorial mistake.  
> **username_0**: Thank you. I thought something was awry at first glance.  
> **username_0**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/105585/discussion-between-tom-and-slawomir-username_1).  

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