# Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself.

In my textbook there's a problem that asks why $R^{2}$ in the regression of $Y$ on $X =$ the sample correlation between X and Y the whole squared.

I've been throwing my head against this for a while and I keep getting stuck because in the correlation coefficient there is a $X$ and $\bar{X}$ term, whilst in the $R^{2}$ term there is no such thing.

Can anyone provide a derivation as to why $R^{2}$ is the correlation coefficient squared?

Thanks!

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It might help if you define the terms in your question. What is the equation for $R^2$, in particular? – Rahul May 10 '12 at 9:27
If by $R^2$ you mean the "explained variance", then stats.SE might be a more suitable site for this question. See, for example, this question or this one for some ideas related to this. – Dilip Sarwate Jan 4 '13 at 23:46

The complete proof of how to derive the coefficient of determination $R^{2}$ from the Squared Pearson Correlation Coefficient between the observed values $y_{i}$ and the fitted values $\hat{y}_{i}$ can be found under the following link:

http://economictheoryblog.wordpress.com/2014/11/05/proof/

In my eyes it should be pretty easy to understand, just follow the single steps.

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There are many forms of the computation available online (such as the Wikipedia page on the correlation coefficient http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient#Pearson.27s_correlation_and_least_squares_regression_analysis ) but note that this is a magical algebraic property of least squares linear regression, not linear regression in general.

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hmm what I don't understand about that is why the correlation coeff equation doesn't have an X term any more? – Scubadiver Apr 10 '12 at 5:49
Which equation does not have an X term? – zyx Apr 10 '12 at 6:03

There are different forms to express R2: Some expressions have (X-Xbar) squared in the numerator, while others express it just with the square of predicted ys. All forms are equivalent.

References: Dougherty; Gujarati; Wooldridge

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