# Imposing diagonality of error covariance matrix when the CAPM holds

Assuming that the CAPM holds, the total risk of an asset can be partitioned into systematic risk (associated with the market factor) and idiosyncratic risk. Idiosyncratic risk is asset specific. Does that imply that the error covariance matrix $$\Sigma_t=\text{Var}(\varepsilon_t)$$ from a cross-sectional regression $$r^*_{i,t}=\alpha_i+\beta_i r^*_{m,t}+\varepsilon_{i,t}$$ is a diagonal matrix?

If so, would it make sense to impose this diagonality when testing the CAPM (e.g. via GMM as discussed in Cochrane "Asset Pricing" (2005) Part II), e.g. when testing $$H_0\colon \alpha_i=0 \ \forall i$$? (I believe the test statistic involves an inverse of $$\hat\Sigma$$ where the latter "covers" all time periods $$t=1,\dots,T$$.) I am interested in this, as I hope it could alleviate the problem of inverting a large unrestricted estimated covariance matrix when the amount of time series observations is quite small.

• The CAPM probably implies that there is a single factor that is priced. However, there may be other factors that that are not priced that make the covariance matrix nondiagonal. This would be impossible in a market consisting of only two assets ($N=2$) but becomes possible in a larger market ($N>2$); see e.g. Petersen (2009) section "Asset Pricing Application" (end og p. 34 of the free version here. Jan 31 at 17:25

No, the CAPM does not imply the error covariance matrix is diagonal. The distinction between systematic and idiosyncratic risk under the CAPM is not as simple as indicated at the start of the question. The CAPM considers expected values, not the covariance matrix. It implies that there is a single factor that is priced, i.e. it has something to say about the expected return. However, there may be other factors that that are not priced that make the covariance matrix nondiagonal.*

*This would be impossible in a market consisting of only two assets ($$N=2$$) but becomes possible in a larger market ($$N>2$$); see e.g. Petersen (2009) section "Asset Pricing Application" (end of p. 34 of the free version here).

• This is correct. A simple example is that even after controlling for the market return, the residual returns $\epsilon$ for Home Depot and Lowes are going to be cross-sectionally correlated: they're in the same industry and exposed to common, industry specific shocks. If Home Depot has a bad quarter, Lowes probably did too. Arguably the biggest statistical problem when dealing with returns is that they're cross-sectionaly correlated. There is some structure in the panel context in that treating error terms from different time periods as orthogonal is generally quite defensible. Feb 15 at 17:00

The notation $$\Sigma$$ is often used for covariance matrix Var($$r_t$$).

Using notation $$\Psi_t = \text{Var}(\epsilon_t)$$, a common modeling assumption is that the residuals are independent, $$\epsilon_{i,t} \perp \epsilon_{j,t}, ~i\neq j$$. Given this independence assumption, the covariance of residual returns matrix $$\Psi$$ is diagonal.

Some factor models (e.g. Barra) implement a concept of "linked specific risk" (LSR), perhaps to model two share classes of the same company. A residual return covariance matrix implementing a concept such as LSR would exhibit non-zero covariance for specific returns.

Idiosyncratic risk is asset specific. Does that imply that the covariance matrix ... is diagonal?

If your model assumes (your hypothesis asserts) idiosyncratic specific risk, the covariance of residual returns is diagonal. Should you collect sufficient data to reject this assumption, as the Barra model does, your residual returns would not be independent, and the residual return covariance matrix would not be diagonal.

would it make sense to impose this diagonality

Starting with a simple (maximum entropy) model "makes sense". Adding complexity under duress (rejecting the current hypothesis based upon data) as the Barra example illustrates makes sense. Modeling is an iterative process. Think deep fakes. If you can spot a feature that distinguishes the "real" from the "fake" image (or return series), impose a constraint that the synthesized return series' feature match the observed return series' feature.

the problem of inverting a large unrestricted estimated covariance matrix

A large estimation effort benefits from regularization. Two potential methods for your modeling efforts are:

• a Bayesian framework, where the prior is a simple model, and the observed data transforms this simple starting point; and,
• a maximum entropy framework, where features are added iteratively to a current model (as you "reject the null hypothesis" in response to statistically significant observed data features inconsistent with synthesized data features).

I wrestled these and related concepts in more detail in my thesis.

Edit: (response to first comment below)
Using Sherman-Morrison-Woodbury, and assuming you have a common factor model that could be expressed as loadings on orthogonal common factor returns (rotate and scale original factor returns if necessary) plus residual returns, the effort to invert the covariance matrix is modest. With a $$k$$-factor model, the only inversion required is of a $$k \times k$$ matrix $$I_k + L^\textrm{T}\Psi^{-1}L$$. The remainder of the process is matrix multiplication and matrix subtraction.

\begin{align} \Sigma &= L L^\textrm{T} + \Psi \\ \Sigma^{-1} &= \Psi^{-1} - \Psi^{-1}L (I_k + L^\textrm{T}\Psi^{-1}L)^{-1}L^\textrm{T} \Psi^{-1} \\ \end{align}

• Thank you! I guess my main concern is whether I am interpreting the CAPM correctly such that it indeed implies a diagonal $\Psi_t$. Regarding inverting a large matrix, I do appreciate your advice. However, for now I would like to stick as closely as possible to the traditional/classical setup for testing asset pricing models as discussed in Cochrane's textbook (say, Fama-MacBeth and also GMM). Within that setup, I wonder if the assumption of diagonality actual helps with inverting the matrix (my intuition says it should). Jan 31 at 14:30
• Look into Sherman-Morrison-Woodbury. It makes inversion of HUGE matrices easy, if they are invertible + low rank modification. A factor model is a low rank modification to the easy to invert diagonal residual-returns matrix. Matrix version is in "Generalization" section of en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula . (your intuition is right!) Jan 31 at 15:27