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

["Tobias Pfaff" <tobias.pfaff@uni-muenster.de>]

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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