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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?

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  • $\begingroup$ 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 $\endgroup$ – mdewey Oct 16 '16 at 11:08
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    $\begingroup$ Hint: $\boldsymbol{\beta}$ is a vector. $\endgroup$ – tchakravarty Oct 16 '16 at 17:04
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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 $\beta$.

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    $\begingroup$ In what sense are the parameters--which are considered fixed--supposed to be a "population"?? $\endgroup$ – whuber Oct 16 '16 at 21:08

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