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 $\text{Cov}(e_i,e_i) = \text{Var}(e_i)$.


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.

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    $\begingroup$ A useful way to interpret covariance, standard deviation, and correlation comes from linear algebra. Covariance is an inner product of two mean zero random variables (it's like the dot product), standard deviation is the norm induced by that inner product, and correlation is the cosine of the angle between the random variables. $\endgroup$ – Matthew Gunn Dec 30 '17 at 6:22

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