Consider the following regressions:
$y_1 = \beta_1 \cdot x + \gamma_1 \cdot \epsilon_1 $
$y_2 = \beta_2 \cdot x + \gamma_2 \cdot \epsilon_2 $
With $\epsilon_1$, $\epsilon_2 \tilde{} N(0,1) $ as usual.
How do you derive the single factor model below?
$y_1 = \sqrt{\rho} \cdot x + \sqrt{1 - \rho} \cdot \epsilon_1 $
$y_2 = \sqrt{\rho} \cdot x + \sqrt{1 - \rho} \cdot \epsilon_2 $
where $\rho$ is the correlation between $y_1$ and $y_2$.
I've looked for a proof of this, but haven't been able to find it anywhere...