Deriving a One-Factor Correlation Model Under Equal Loadings

Summary

The document derives a two-variable single-factor model from regressions in which both variables share a common factor and each has an idiosyncratic error. The derivation requires equal factor loadings and equal residual coefficients for the two variables. It also assumes the residuals are mutually uncorrelated and each residual is uncorrelated with the common factor.

With both variables normalized to have zero mean and unit variance, their correlation equals their covariance. Under these conditions, the covariance between the variables comes from their shared factor, while each variable’s variance combines the common-factor and residual contributions. The resulting correlation can therefore be represented by the shared factor’s variance contribution, giving the square-root coefficients in the stated model. This is a result for the specified symmetric assumptions; unequal loadings, correlated residuals, or different scaling would require a different derivation.

Key ideas

• Equal factor loadings and residual coefficients are required for the symmetric model shown.
• The residuals must be uncorrelated with one another and with the common factor.
• Unit variance normalization makes covariance equal to correlation.
• The shared factor accounts for cross-variable covariance under these assumptions.

Tags

# Deriving the single factor model


# Deriving the single factor model












Consider the following regressions, with the common factor $x$:

$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, and $Cov(x, \epsilon_j) = 0$, for all $i$.

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 by J4y (score 0, accepted)

https://quant.stackexchange.com/a/24455

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.