Factor model assumptions

Question

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:

1. $\epsilon = (\epsilon_1, \ldots, \epsilon_d)'$ is a random vector of idiosyncratic terms, which are uncorrelated and have mean zero
2. $\text{cov}(F, \epsilon) = 0$.

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 1

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 2

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.