How Independent Residuals Affect Asset Covariance and Correlation
Summary
The document asks why adding regression residuals to asset returns can leave their covariance unchanged. In a factor model, return covariance separates into covariance explained by the shared factors and covariance between residuals. If residuals are uncorrelated with the factors and with one another across different assets, they add no off-diagonal covariance; each asset’s own residual variance contributes to the diagonal of the covariance matrix.
The answer highlights that covariance and correlation respond differently. Independent noise can increase each asset’s variance while leaving the covariance between two assets unchanged, so their correlation can fall. This resolves the intuition that noisy versions of perfectly correlated assets should become less correlated. The explanation is conceptual rather than a worked derivation, and the result depends on the stated uncorrelated-error assumptions; correlated residuals can change cross-asset covariance.
Key ideas
- Uncorrelated residuals across different assets do not add to their covariance.
- Each asset’s residual variance increases its own variance and appears on the covariance matrix diagonal.
- Independent noise can lower correlation even when the covariance between assets is unchanged.
- The conclusion depends on residuals being uncorrelated with factors and with other assets’ residuals.
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# Factor Models: uncorrelated errors don't impact covariances of assets
# Factor Models: uncorrelated errors don't impact covariances of assets
This question stems from time series factor models (e.g., CAPM, Fama-French, etc.), but is a broader idea.
I am trying to comprehend how adding noise to a time series (e.g., error/residual from a regression) doesn't change the covariance between two assets.
For example, let's take two assets that are perfectly correlated with one another. If I add some random noise to each of those two assets (in which the random noise is uncorrelated to the assets and the other random noise), the properties of covariance would say that has no impact on the covariance of the assets, but that just doesn't seem correct.
What am I missing? Is there a more intuitive way of thinking about this?
As it relates to factor models, this essentially shows up in the fact that the asset covariance matrix only receives an addition of the error term variance along the diagonal. As an equation, this shows up in the last term of the following two-factor model:
\begin{eqnarray} \text{Cov}(r_i,r_j) & = & b_{i,1} b_{j,1} \text{Cov}(f_1,f_1) + b_{i,1} b_{j,2} \text{Cov}(f_1,f_2) + b_{i,2} b_{j,1} \text{Cov}(f_2,f_1)\\ & & + b_{i,2} b_{j,2} \text{Cov}(f_2,f_2) + \text{Cov}(e_i,e_j) \end{eqnarray}
Where $\text{Cov}(e_i,e_j) = 0$ unless $i=j$, in which case $\text{Cov}(e_i,e_i) = \text{Var}(e_i)$.
## Answer by Parallax (score 0)
https://quant.stackexchange.com/a/37537
Actually...I think I may have solved my own question. The covariance will stay the same, but the correlation may drastically change. It's hard (or next to impossible) to visualize covariance whereas correlation is fairly easy.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.