TITLE: Coefficient of determination, why?
QUESTION [2 upvotes]: I mean it is written in a book "Statistics for Management and Economics", that coefficient of determination is coefficient of correlation squared. Well, am I the only one to whom this is surprising fact as he expected something more clear or natural?? I mean, if someone can present me the proof why, or why some other, more natural things do not work, like I don't know, absolute value of the coefficient of correlation or something similar to Chebyshev theorem ($1-\text{coefficient of correlation}$)?

REPLY [2 votes]: Suppose you have $n$ paired observations $(x_i,y_i)$ on $(x,y)$ and you want to predict $y$ on the basis of $x$. 
You consider the prediction model $$y=\phi(x)+e$$ where $\phi$ is  the part of $y$ explained by $x$ through $\phi$ and $e$ is the unexplained part.
Suppose you see from the scatter plot of $x$ and $y$ that $\phi$ is more or less linear. 
So you choose $$\phi(x)=a+bx$$
Then the least square linear predictor of $y$ obtained on the basis of $x$ is $$\hat y=\hat a+\hat b x$$, where
$$\hat a=\bar y-\hat b\bar x\quad,\quad \hat b=\frac{\operatorname{cov}(x,y)}{\operatorname{var}(x)}$$
It can be shown that
\begin{align}
\operatorname{var}(\hat y)&=\operatorname{var}(\hat a+\hat b x)
\\&=\hat b^2\operatorname{var}(x)
\\&=r^2 \operatorname{var}(y)
\end{align}
, where $r$ is the correlation coefficient between $x$ and $y$.

A measure of efficacy of the predictor $\phi$ is given by the proportion of variation in $y$ explained by $\phi$, i.e., $$\frac{\operatorname{var}(\hat y)}{\operatorname{var}(y)}=r^2$$, which is termed as coefficient of determination.

Of course, the coefficient of determination is numerically equal to the square of the correlation coefficient, but that is hardly a definition or a motivation for the former.
When $r^2=0$ the linear prediction of $y$ obtained on the basis of $x$ is worst, and when $r^2=1$ the prediction  is perfect as $\phi$ explains the variability in $y$ completely.
For more details, the following threads might be helpful: 

Correlation Coefficient and Determination Coefficient 
https://stats.stackexchange.com/questions/123651/geometric-interpretation-of-multiple-correlation-coefficient-r-and-coefficient?noredirect=1&lq=1
https://stats.stackexchange.com/questions/1447/coefficient-of-determination-r2-i-have-never-fully-grasped-the-interpretat?noredirect=1&lq=1.
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