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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...

Silverfish
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An old man in the sea.
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    Why don't you post your proof up until the point where you get confused – Steve S Aug 12 '14 at 23:48
  • 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. – whuber Aug 13 '14 at 00:19
  • @SteveS I've edited my post. – An old man in the sea. Aug 13 '14 at 00:20
  • @whuber I've managed to find only two posts, and only one had an answer from what I could understand... http://stats.stackexchange.com/questions/99669/the-equivalence-of-sample-correlation-and-r-statistic-for-simple-linear-regressi/99677#99677

    This second link, even though it asks the same question among others, it didn't received an answer for this question. http://stats.stackexchange.com/questions/32294/regression-r2-and-correlations

    If there's an answer already on the CV, I would be very thankful if you could direct me towards it.

     – An old man in the sea. Aug 13 '14 at 00:28
  • @whuber Also, most answers on similar questions do not prove this. The answers remain at an intuitive (word) level. – An old man in the sea. Aug 13 '14 at 00:33
  • [Back to the drawing board...] There are two things to keep in mind: 1.) $\bar{y} = a + b\bar{x}$ and 2.) $b = \frac{cov(x,y)}{\sigma_x^2} = \frac{\sigma_y}{\sigma_x}*\rho(x,y)$ – Steve S Aug 13 '14 at 01:39
  • Since this is routine bookwork, would you mind adding the self-study tag (and check its tag wiki info for how such questions are handled) – Glen_b Aug 13 '14 at 04:21
  • @Glen_b It's been added. – An old man in the sea. Aug 13 '14 at 09:49
  • @whuber Is your answer also valid for multiple regressions? – An old man in the sea. Aug 13 '14 at 09:56
  • On a first read through, it looks to me like whuber's argument carries directly over to multiple regression; I didn't spot any obvious gaps that needed filling. It's possible I missed something though. – Glen_b Aug 13 '14 at 10:48
  • @Glen_b and could please tell me what you think of this answer of mine?Is it correct? http://stats.stackexchange.com/questions/70190/proving-that-r2-is-r2/111707#111707 – An old man in the sea. Aug 13 '14 at 12:43
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    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) – whuber Aug 13 '14 at 14:04

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