# What does the asymptotic heteroskedastic variance-covariance matrix of beta look like?

I don't really understand the variance-covariance matrix of beta when there is heteroscedasticity. Can someone explain what the matrix looks like? I am talking about:
$$V_\beta=Q_{xx}^{-1}\Omega Q_{xx}^{-1}$$

where,

• $V_\beta$ is the variance of beta (asymptotic, so equal to the population variance for large n).
• $Q_{xx}$ is $X'X$ and $\Omega$ is the variance of $X'e$ where the estimator for $\beta$ is $(X'X)^{-1}X'y$ and $e=y-X\beta$. Therefore $y=e+X\beta$ is substituted for $Y$ and then solved for $\hat\beta-\beta$, getting $\hat\beta-\beta=(X'X)^{-1}X'e$.

So if I understand correctly, the variance-covariance matrix shows the variance on the diagonal of $\beta$, but what are on the off diagonal elements?

• Can you expand on what the terms in your formula stand for? Telling us how far you have got so far in your thinking would also be useful – mdewey Oct 16 '16 at 11:08
• Hint: $\boldsymbol{\beta}$ is a vector. – tchakravarty Oct 16 '16 at 17:04

As tchakravarty noticed, $\beta$ is a column vector of all the parameters of the model. So the variance-covariance matrix shows on the diagonals the covariance of the $\beta$ with itself (=variance), and on the off diagonal elements it shows the covariance of individual $\beta$'s with other $\beta$'s.
In conclusion shows this matrix the asymptotic heteroskedastic variance of the estimator $\hat \beta$, which converges by the Central Limit Theorem (CLT) and the Weak Law of Large Numbers (WLLN) to the variance of the population $\beta$.