# Understanding General Least Squares

From what I understand, we have our OLS estimator in matrix form which is,

$$\beta_{OLS} = (X'X)^{-1}X'Y$$

What we want to do is transform this, as the assumption that we have constant variance is not met. So we take a vector, p, for which,

$$\Omega = pp'$$

$$\Omega= \begin{bmatrix}\sigma_1^2&0&...&0\\0&\sigma_2^2&...&\vdots\\\vdots&\vdots&\ddots&0\\0&...&0&\sigma_n^2\end{bmatrix}$$

The we use this to transform $$\beta_{OLS}$$ to get $$\beta_{GLS}$$ :

$$\beta_{GLS} = (X'P^{-1}P^{-1}X)^{-1}X'P'^{-1}P'^{-1}P'y$$

My question is one to do with understanding, I don't get what p intrisically is, I get that it is being used to transform our estimate to obtain a variance term which meets the GM assumptions, but how would one actually go about finding what p is in a paractical sense?

Is the point of the $$\Omega$$ matrix to show what we are working to find, i.e. the actual variances?

Thank you.

• Could you clarify the question a little more? Are you asking for an explicit way to estimate the matrix $\Omega$? – Walrasian Auctioneer Apr 16 '20 at 22:18