Conceptual explanation of regression coefficient?

Question

I understand, mathematically, how to get the estimates of coefficients in ordinary least squares. What I am struggling with is coming up with a conceptual, geometric explanation for the correlation part of the simple linear regression coefficient.

Say we have the following equation: $$ y = \hat\beta_1x + \hat\beta_0 $$ where $$ \hat\beta_1 = \frac{S_{xy}}{S^2_x} = r_{xy}\frac{S_y}{S_x} $$ and $$ \hat\beta_0 = \overline{y} - \hat\beta_1\overline{x} $$

If I understand correctly, we are essentially just getting estimates of spread ($\hat\beta_1$) and position ($\hat\beta_0$) shifts that can turn $x$ into $y$ as best as possible (least squares). So from what I understand, the ratio of standard deviations ($\frac{S_y}{S_x}$) makes it so $x$ has the same spread as $y$, and then the estimate of $\beta_0$ moves $x$ to the same position as $y$.

What I am missing is what the correlation $r_{xy}$ is doing in this linear transformation. If we have shifted the spread and position of $x$ as above, what is $r_{xy}$ representing?

Answer 1

$r_{xy}$ is the slope of the OLS regression line in that transformed space. You can also think of $r^2_{xy}$ as the proportion of the variance in the transformed $y$ variable that is explainable by virtue of knowledge of the transformed $x$, and $1-r^2_{xy}$ as the error variance, or the variance that cannot be so explained.