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Suppose I have panel data $\{y_{it},X_{it}\}_{i=1...N,t=1...T}$ and the following linear (seemingly unrelated regression) model:

$y_{it} = \beta_i X_{it} + u_{it}$

where the errors are correlated across $i$ but not $t$. That is, defining $u_t = [u_{1t} \; u_{2t} \;... \; u_{Nt}]'$, $u_t \sim \mathcal{N}(0,\Sigma)$.

Suppose I obtain estimates $\hat{\beta}_i$ and residuals $\hat{u}_{it}$ using feasible GLS and then recover the covariance matrix of the residuals, $\hat{\Sigma}$. It's easy to find an asymptotic theory of $\hat{\beta}_i$ but my question is about $\hat{\Sigma}$:

I know that $\hat{\Sigma}$ is, for instance, consistent. But do we know what its asymptotic (or, fingers crossed, finite sample) distribution is? Like, if I wanted to test that two arbitrary elements of $\hat{\Sigma}$ were equal, do we have a way to do this?

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  • $\begingroup$ Asymptotically, it converges to normal, but the properties of converge is bad. Likelihood ratio test can be used, which is better than tests based on asymptotic normal. $\endgroup$ – user158565 Jul 24 at 16:05

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