# How to derive the distribution of OLS starting from the sample moments?

I know I am supposed to start from

$$N^{1/2}[N^{-1}\sum x_{i}u_{i}]$$

Then by central limit theorem that that it is asymptotically

$$N(E(x_{i}u_{i}),var(x_{i}u_{i}))$$

and $$E(x_{i}u_{i})=0$$

so $$N(0,var(x_{i}u_{i}))$$

from there I cant see how I get to the correct result which is supposed to be

$$N(0,(E(x_{i}x'_{i}))^{-1}E(u^2_{i}x_{i}x'_{i})(E(x_{i}x'_{i}))^{-1})$$

• it looks like you have OLS with heteroskedastic error term so check out the well known paper by Halbert White whose title currently escapes me. I'll see if I can find the name of it. – mlofton Dec 20 '18 at 16:25
• The derivation is probably in here. pdfs.semanticscholar.org/f9cc/… – mlofton Dec 20 '18 at 16:26
• Here's another link that can probably lead to some good references. en.wikipedia.org/wiki/… – mlofton Dec 20 '18 at 16:30