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I'm having a hard time proving that $R^2$ is equal to the square of the sample correlation between $Y$ and $\hat{Y}$. Every book I search tells me that's very easy, like verbeek. They just state that from the definition of $R^2$ and knowing that $SST=SSR+SSE$, it's very easy to prove the claim. However, I've spent a lot of time thinking about it, with no sucess.

Any help would be appreciated.

EDIT (My try): $R^2=\frac{SSE}{SST}=1-\frac{SSR}{SST}$

Sample correlation$^2=\frac{\left(\sum(\hat{y_i}-\bar{y})(y_i-\bar{y})\right)^2}{ SST\cdot SSE }$

From then on, I have no idea...

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    $\begingroup$ Why don't you post your proof up until the point where you get confused $\endgroup$ – Steve S Aug 12 '14 at 23:48
  • $\begingroup$ I believe you would prefer to prove that $R^2$ is the square of that correlation. A little time spent searching our site will uncover several demonstrations of this and similar results. $\endgroup$ – whuber Aug 13 '14 at 0:19
  • $\begingroup$ @SteveS I've edited my post. $\endgroup$ – An old man in the sea. Aug 13 '14 at 0:20
  • $\begingroup$ @whuber I've managed to find only two posts, and only one had an answer from what I could understand... stats.stackexchange.com/questions/99669/… This second link, even though it asks the same question among others, it didn't received an answer for this question. stats.stackexchange.com/questions/32294/… If there's an answer already on the CV, I would be very thankful if you could direct me towards it. $\endgroup$ – An old man in the sea. Aug 13 '14 at 0:28
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    $\begingroup$ Yes, it is valid for multiple regressions: none of the reasons adduced to support that proof refers to or implicitly relies on there being a single independent variable. A minor variation of the simulation presented there (to extend it to multiple regression) helps confirm this result. (cc @Glen_b) $\endgroup$ – whuber Aug 13 '14 at 14:04