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I'm studying Econometrics from Stock and Watson, and when mentioning the F-statistic in the Multiple Regression (Chapter 7), there is this formula:

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How can I estimate $\hat{\rho}_{t_{1},t_{2}}?$ Can I use the correlation coefficient between the two variables?

I suppose there is a standard formula used for the statistical softwares, but I could not find.

User1865345
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Oalvinegro
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  • While SW's exposition makes pedogogical sense to discuss how the F test works, you do not really need to estimate this expression - I believe it is more fruitful to proceed from the general expression of the F statistic directly, as discussed e.g. here: https://stats.stackexchange.com/questions/258461/proof-that-f-statistic-follows-f-distribution/258476#258476 – Christoph Hanck Feb 22 '23 at 11:03
  • See also the page from Christoph Hanck's book. – User1865345 Feb 22 '23 at 13:00
  • @ChristophHanck, to be frank, we were never bothered with expressing the F statistic in terms of individual $t$ statistic. Even though the authors provided a heuristic argument, they didn't provide a derivation. Are there any derivation in the same vein? I didn't didn't find any other books expressing the F statistic in this way. – User1865345 Feb 22 '23 at 13:08
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    I have been pushing around the expression for the F statistic for a bit but have also not obtained something which would exactly match S&W's (7.9) so far... – Christoph Hanck Feb 27 '23 at 08:27
  • @ChristophHanck do we need this expression to construct the confidence set mentioned in the same section of the book? – Oalvinegro Feb 27 '23 at 15:28
  • No, as I suggested, we can work with a general expression for the F statistic. S&W's expression is just an alternative way to write the statistic (albeit one we do not understand atm :-) ) – Christoph Hanck Feb 27 '23 at 16:02

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