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Let's assume that we want to obtain the coefficients of the following bivariate regression: $Y=\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$

However, we don't have access to the data $(X_1,X_2,Y)$. The only information available are:

  • Coefficients of the univariate regression $Y=\gamma_0 + \gamma_1 X_1 + \varepsilon$
  • Coefficients of the univariate regression $Y=\lambda_0 + \lambda_1 X_2 + \nu$
  • Variances of $X_1$ and $X_2$, plus correlation between these two
  • Variance of $Y$

How can we obtain $\beta_1$ and $\beta_2$ from this information?

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    $\begingroup$ I’m voting to close this question because it is not specific to Quantitative Finance and belongs on Cross Validated Stack Exchange instead. $\endgroup$ Apr 5 at 8:10

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Betas are $(x'x)^{-1}(x'y)$. You know $(x'x)^{-1}$ from variances and correlation. You know $x'y$ from the correlation between each x and y (imply from regression coefficient).

Edit for clarity, x is a matrix of the data of the final regression.

Final edit: If you know the covariance matrix of x1,x2 and y, then you know the joint distribution completely and therefore you know the new coefficients completely. They are in this case, deterministic and one need not calculate the coefficients to realise this. This is ofc assuming normality

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  • $\begingroup$ I spent some time trying to hash out the details of what you wrote above and was not able to compute the terms. Could you provide more details on how you obtain the numerator and denominator. Then, coincidentally,, I pointed to this question from cross validated because there was a similar question over there. Someone replied that above is incorrect but I'd like to know for sure. Thanks a lot for any details. stats.stackexchange.com/questions/644458/… $\endgroup$
    – mark leeds
    Apr 7 at 18:08
  • $\begingroup$ @markleeds that is a different question that also cleaves y. What I've written is by definition correct, can you tell me where you were stuck? $\endgroup$
    – Arshdeep
    Apr 7 at 21:23
  • $\begingroup$ Hi. The guy that claimed incorrect definitely read the question and answer and has a gigantic rating on cross validated. So, that's why I asked. I got stuck calculating $x^{\prime } y$. Thanks for any details. $\endgroup$
    – mark leeds
    Apr 7 at 22:29
  • $\begingroup$ x'y becomes ([y_mean product(x1,y) product(x2,y)]/n) and covariances are known from the other regressions. WLOG means of each variable are 0 so products can be related to covariances. I am 100% confident that the OLS estimator can be written as the way I have, by every definition, it shouldn't be hard to google it. $\endgroup$
    – Arshdeep
    Apr 7 at 23:48
  • $\begingroup$ @markleeds please see edit, would be happy to see someone generate multiple possibilities of the betas as was done in the link you pasted. Thanks! $\endgroup$
    – Arshdeep
    Apr 8 at 0:05

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