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...


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.