Interpreting Orthogonality and Residual Assumptions in Factor Models
Summary
The document examines a linear factor model in which an observed vector is represented by an intercept, factor exposures, and residual terms. It asks how to interpret the condition that factors are uncorrelated with residuals, and whether mutually uncorrelated residuals imply that all systematic risk factors have been captured.
The accepted response explains factor-residual orthogonality as a consequence of regression projection: the fitted component lies in the factor-spanned space and the residual lies in its orthogonal complement. By contrast, residuals need not be mutually uncorrelated; treating them that way is a simplifying assumption that may be acceptable when factors explain most variance, though correlated residual structure can exist. Another response emphasizes that factor-residual uncorrelatedness helps split covariance into factor and residual components. The discussion distinguishes regression geometry from extra modeling assumptions, but does not supply empirical tests for those assumptions.
Key ideas
- Regression projection makes the residual component orthogonal to the included factors under the stated setup.
- Orthogonality between factors and residuals is distinct from assuming residuals are mutually uncorrelated.
- Residual components may be correlated with one another even when they are orthogonal to the factor space.
- Assuming uncorrelated residuals simplifies covariance decomposition into factor and residual parts.
- The model’s assumptions do not by themselves prove that every systematic risk factor has been included.
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Full text
# Factor model assumptions
# Factor model assumptions
I was reading on Factor Models in the book Quantitative Risk Management by McNeil, et al. In section 3.4.1 they introduce a linear factor model $$X = a + BF + \epsilon,$$ where $X \in R^d$, $F \in R^p$. They make the following assumptions:
- $\epsilon = (\epsilon_1, \ldots, \epsilon_d)'$ is a random vector of idiosyncratic terms, which are uncorrelated and have mean zero
- $\text{cov}(F, \epsilon) = 0$.
I understand that if there were an additional factor, say $B_{p+1} F_{p+1}$, then $\epsilon$ would no longer be uncorrelated. So perhaps the Assumption (1) makes sure that all the risk factors have been captured and the remaining randomness is truly idiosyncratic.
How do we understand Assumption (2)?
## Answer by Hans (score 3, accepted)
https://quant.stackexchange.com/a/14852
As a matter of fact, Assumption 2 is natural. It is Assumption 1 that needs justification.
Strictly speaking Assumption 2 is not an assumption. It is simply a corollary of the regression of $X$ against $F$. In the language of linear algebra, it is the decomposition of vector space of all $X$ into the subspace spanned by $F$ and its orthogonally complementary subspace. This equation is exact.
However, Assumption 1 is not necessarily satisfied except the part about mean zero which is simply the result and purpose of having constant $a$ absorbing all nonzero means. One can easily construct any number of vectors with their complementary component highly correlated with each other. However, presumably most of the variance is captured by $F$, and so whatever you may say about the residual vector is immaterial except perhaps their residual variance. Therefore you may as well treat them as if they are completely uncorrelated.
## Answer by John (score 3)
https://quant.stackexchange.com/a/14851
The assumptions from factor models tend to be similar to the assumptions that you see in regression.
The first assumption is pretty straightforward, errors are uncorrelated with mean zero. I wouldn't say that it necessarily means that all risk factors have been captured. However, for the purposes of using the factor model, it is basically assuming that there are no more systemic factors.
The second assumption means that the factors and the errors are uncorrelated. The main reason for this assumption is that it facilitates being able to split up the covariance so that there is a factor component and a residual component.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.