Deriving the single factor model

Question

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...

Answer 1

After some digging arround, I found that you can derive the single factor model under the following assumptions:

$\beta_1 = \beta_2$

$\gamma_1 = \gamma_2$.

In addition, we require $Cov(\epsilon_1, \epsilon_2) = 0$ and $Cov(\epsilon_i, x) = 0$ for $i = 1, 2$.

Finally, we would require $y_1$ and $y_2$ to be normalized $\tilde{} N(0, 1)$ for the correlation to be equal to the co-variance.

This way:

$\sigma_i^2 = \beta^2 + \gamma^2$

$\sigma_{i,j} = \beta^2$

and we can derive the above.