Pearson multiple correlation upper bound Suppose we have multiple independent variables y1,y2,...  along with their Pearson correlations r1, r2,... related to a dependent variable x. (Assume that we do not know the correlations between the independent variables).
Then the lower bound of the multiple correlation is (AFAIK) the largest of the correlations absolute values.
What is the upper bound of the multiple correlation in terms of r1, r2,... ?
Thanks
 A: Let's start with two variables.
Suppose $\phi_1, \phi_2, \eta$ are uncorrelated variables, each with mean 0 and variance 1.
Set $y_1 = \alpha_1 \phi_1 + \eta$ ,$y_2 = \alpha_2 \phi_2 - \eta$, and $x = \alpha_1 \phi_1 + \alpha_2 \phi_2 = y_1 + y_2$.
Then $E\left[y_1^2\right] = 1 + \alpha_1^2$, $E\left[y_2^2\right] = 1 + \alpha_2^2$, $E\left[x^2\right] = \alpha_1^2 + \alpha_2^2$, $E\left[y_1 x\right] = \alpha_1^2$, and $E\left[y_2 x\right] = \alpha_2^2$.
From this we have that
$\mbox{corr}(y_1, x) = \frac{\alpha_1^2}{\sqrt{1 + \alpha_1^2}\sqrt{\alpha_1^2 + \alpha_2^2}}$
and 
$\mbox{corr}(y_2, x) = \frac{\alpha_2^2}{\sqrt{1 + \alpha_2^2}\sqrt{\alpha_1^2 + \alpha_2^2}}$
Clearly, the multiple correlation is 1 (as $x = y_1 + y_2$), but looking at the above two equations, they are solvable for $r_1$ and $r_2$. So it doesn't look there is a nontrivial upper bound.
I think this can be extended to an arbitrary number of variables (by making each $y_i$ include $\frac{1}{n - 1} \eta$ for ($i = 1, ..., n -1$), and $y_n$ include $-\eta$), but the terms got a bit messy.
