Correlation Coefficient and Determination Coefficient

Question

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

In my textbook there's a problem that asks "why is $R^{2}$ in the regression of $Y$ on $X$ equal to the square of the sample correlation between X and Y?"

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!

Answer 1

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.

Answer 2

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