In linear regression we have the delightful result that if we fit the model

$E[Y] = \beta_1 X_1 + \beta_2 X_2 + c$

then

$R^2 = \mathrm{Cor}(Y,X_1) \beta_1 + \mathrm{Cor}(Y, X_2) \beta_2$

The only proof I know is not in anyway constructive or insightful (see below), and yet to look at it it feels like it should be readily understandable.

Example thoughts:

• The $\beta_1$ and $\beta_2$ parameters give us the 'proportion' of $X_1$ and $X_2$ in $Y$, and so we are taking respective proportions of their correlations...
• The $\beta$s are partial correlations, $R^2$ is the squared multiple correlation... correlations multiplied by partial correlations...
• If we orthogonalize first then the $\beta$s will be $\mathrm{Cov}/\mathrm{Var}$... does this result make some geometric sense?

None of these threads seem to lead anywhere for me. Can anyone provide a clear explanation of how to understand this result.

Unsatisfying Proof

If we standardize and centre the $Y$, $X_1$ and $X_2$ data then

$R^2 = \frac{SS_{reg}}{SS_{Tot}} = SS_{reg} = <(\beta_1 X_1 + \beta_2 X_2)^2> = <\beta_1^2 X_1^2> + <\beta_2^2 X_2^2> + 2<\beta_1\beta_2X_1X_2>$

and

$\mathrm{Cor}(Y,X_1) \beta_1 + \mathrm{Cor}(Y, X_2) \beta_2 = <YX_1>\beta_1 + <Y X_2> \beta_2$

$=<\beta_1 X_1^2 + \beta_2 X_1 X_2> \beta_1 + <\beta_1 X_1 X_2 + \beta_2 X_2^2> \beta_2$

$=<\beta_1^2 X_1^2> + <\beta_2^2 X_2^2 > + 2<\beta_1 \beta_2 X_1 X_2>$

QED