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We were told, that under the assumption that $y=\mathbf{x}'\beta_\circ+u$ in the representation of the ordinary least squares estimator (with $N$ being the sample number) $$\hat{\beta}=\beta_\circ + (1/N \cdot\mathbf{X'X})^{-1} (1/N \cdot\mathbf{X'u})$$ the $(1/N \cdot\mathbf{X'X})^{-1}$ converges to the expectation $\mathbb{E}(\mathbf{xx}')$ (and the $ (1/N \cdot\mathbf{X'u})$ to $\mathbb{E}(\mathbf{x}u)$), because of the LLN.

Why is that like that, how can $\mathbb{E}(\mathbf{xx}')$ be seen to be the expected value?

Jaleks
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  • Hi: I'm pretty sure that what you have written has a typo or two in it. Do you have access to Fumio Hayashi's text called "econometrics" ? This text does a really nice job of explaining what the assumptions are for the finite sample case and asymptotic case and gives all of the gory details of the derivations. I couldn't do it justice if I attempted to explain any of it. I highly recommend taking a look at this text. – mark leeds Jul 24 '22 at 16:10
  • my mistake because Hayashi's text, although quite rigorous, does not include the case of random regressors which is what your case is. So, I'm changing my text recommendation. It's best to check out "An Introduction To Econometric Theory" by James Davidson. He covers the case of random regressors in the later chapters, namely 12 and 13. Your specific question is detailed on page 201 in section 13.5 but it's best to read the chapters completely. – mark leeds Jul 24 '22 at 16:58
  • unfortunately I have neither the one nor the other of that expensive text books available, thanks for your thoughts nevertheless – Jaleks Jul 25 '22 at 08:08
  • Hi Jalek: Look at the early pages ( I would start from the very beginning ) of this and particularly pages 10-13. https://www.bauer.uh.edu/rsusmel/phd/ec1-7.pdf I hope it helps. – mark leeds Jul 26 '22 at 15:25
  • Unfortunately I could not find anything to Exx' there? – Jaleks Aug 03 '22 at 14:13
  • let me print out and take a look. I thought it was there but I could be mistaken. – mark leeds Aug 03 '22 at 17:40
  • Hi: Look on page 13. the title of the page is "consistency of ols: b". It is shown there that b is unbiased given the assumptions. – mark leeds Aug 03 '22 at 17:45

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