# Few-clusters bias correction for cluster robust covariance matrix in random effects model

I'm currently working on some experimental data. Subjects are randomly assigned to one of two treatments. For each treatment I ran three sessions with 20 subjects each. In each session, participants were asked to make a sequence of decisions.

I estimated the effect of the treatment with a model that includes individual random effects. Afterwards, I estimated the cluster robust covariance matrix (CRCME): $$\hat{V}_{CRCME}=(X^{'}X)^{-1}\Bigg(\sum_{s=1}^{S}{X_{s}^{'}\hat{\epsilon}_{s}\hat{\epsilon}_{s}^{'}X_{s}}\Bigg)(X^{'}X)^{-1},$$ where $s$ defines the cluster. This procedure provides a consistent estimator of the covariance matrix which can typically be biased when the number of cluster is little. In my case I have only 6 clusters, so I decided to try the residual correction proposed by Bell and McCaffrey (2002). Instead of the OLS residuals $\hat{\epsilon}_{s}$, they propose $$\tilde{e}_{s}=\sqrt{(S-1)/S}[I_{N_{s}}-H_{ss}]^{-1}\hat{\epsilon}_{s},$$ where $H_{ss}=X_{s}(X^{'}X)^{-1}X_{s}^{'}$. Given that I estimated a random effects model, following McCaffrey, Bell and Botts (2002), I applied this correction on the transformed data: $X^{*}=W^{1/2}X$ and $\hat{\epsilon}^{*}=W^{1/2}\hat{\epsilon}$ where $W^{1/2}$ is the standard weight matrix used to estimate individual random effects models.

The standard errors I obtain with Bell and McCaffrey's correction are bigger than the cluster robust ones, except for the treatment variable and the intercept. Should I worry about that? The treatment variable is the main variable of interest. Do you have any idea about the possible reasons?

References:

• Bell, R.M., and D.F. McCaffrey (2002), "Bias Reduction in Standard Errors for Linear Regression with Multi-Stage Samples," Survey Methodology, 169-179.
• McCaffrey, D.F., Bell, R.M., and C.H. Botts (2001), "Generalizations of bias Reduced Linearization," Proceedings of the Survey Research Methods Section, American Statistical
Association
• Please provide references. Of course the standard errors will get bigger (and that's a step in the right direction as "robust" standard errors are known to be biased down): you essentially have a matrix term $(1-h)^{-1}, h>0$ in the definition of $\tilde e_s$ to inflate residuals by their leverage, of sorts. – StasK Sep 22 '15 at 3:07
• I think you overlooked one important part of my question. The standard errors will get bigger as expected, but not for all variables. It turns out that the standard errors for the treatment variable (1 if treatment, 0 otherwise) will get way smaller. I don't think that this is specific to my data, rather I think that this happens because the treatment variable is a time-invariant dummy. I obtained the same result using different data. – sciacallojo Sep 22 '15 at 8:57