# Inference on Error Covariance Matrix in SUR/GLS?

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?

• 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. – user158565 Jul 24 at 16:05