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Re: st: Comparison of robust and cluster-robust standard errors when the number of clusters is small


From   Austin Nichols <austinnichols@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Comparison of robust and cluster-robust standard errors when the number of clusters is small
Date   Fri, 4 Jan 2013 10:35:55 -0500

Tobias Pfaff <tobias.pfaff@uni-muenster.de>:
As the FAQ says,
"See the manual entries [R] regress (back of Methods and Formulas),
[P] _robust (the beginning of the entry), and [SVY] variance
estimation for more details."
and see an article referenced there (and by
http://repec.org/usug2007/crse.pdf
as well):
http://www.stata.com/support/faqs/stat/stb13_rogers.pdf
--Rogers 1993 is still the best intro, though the commands are all obsolete.
Also see Kish 1965
http://www.amazon.com/Survey-Sampling-Wiley-Classics-Library/dp/0471109495
for an early explanation of one way to measure [possibly negative]
intracluster correlation of x*e, using a quantity which Kish calls roh
[sic].

On Thu, Jan 3, 2013 at 11:43 AM, Tobias Pfaff
<tobias.pfaff@uni-muenster.de> wrote:
> Hi all,
>
> The Stata FAQ explains nicely why cluster-robust standard errors
> (-vce(cluster clustvar)-) can be smaller than robust standard errors
> (-vce(robust)-):
> http://www.stata.com/support/faqs/statistics/standard-errors-and-vce-cluster
> -option/
>
> The FAQ's answer is negative correlation within cluster.
> But could it be that in cases with small number of clusters this answer is
> not sufficient?
>
> Consider a setting with a small number of clusters (in my case 12 clusters)
> and the following standard errors:
>
> Ordinary (OLS) SE: .1109
> Robust SE: .1268
> Cluster-robust SE: .0414
>
> The literature says that an insufficient number of clusters (approximately
> less than 50) can lead to standard errors that are downward biased (e.g.,
> Cameron et al. 2008).
>
> Is it correct to say that my cluster-robust SE is smaller than the robust SE
> due to negative correlation within cluster OR due to downward bias of the
> cluster-robust SE in the case with few clusters?
>
> If the statement is correct, can I find out if one of the reasons can be
> ruled out? Can I measure the negative correlation? Or can I measure the
> downward bias due to few clusters?
>
> In this regard, maybe you guys have a hint for me why (mathematically) the
> SE are downward biased in the case with few clusters? I didn't find an
> answer so far in the literature.
>
> Any comments are appreciated!
>
> Thanks very much,
> Tobias
>
> Literature cited:
> Cameron, Gelbach, Miller (2008), Bootstrap-Based Improvements for Inference
> with Clustered Errors. The Review of Economics and Statistics, 90 (3),
> 414-427.
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