Shanken Correction Dimensions for Multifactor Asset-Pricing Models
Summary
The document addresses two questions about the Shanken correction for errors-in-variables in asset-pricing tests. First, it asks how the correction's covariance terms can be added when the factor covariance is a matrix, and whether multifactor models require the covariance matrix of all factors. Second, it asks why one expression includes an extra factor-covariance term relative to a formula presented in an asset-pricing textbook.
The answer resolves the dimensional concern by distinguishing single-factor from multifactor cases. With one factor, the relevant beta-based term and factor variance are scalars; with multiple factors, the beta-based term is a matrix, as is the factor covariance matrix, so their dimensions match. For the second formula, the response points to a separate summary and notes that the textbook treatment may omit details; it does not fully derive or explain the difference. The discussion clarifies matrix dimensions but leaves the formula comparison and implementation details open.
Key ideas
- In the single-factor case, the correction terms and factor variance are scalars.
- In a multifactor model, the beta-based correction and factor covariance are matrices of compatible dimensions.
- The factor covariance matrix represents the covariance across all included factors.
- The answer does not derive the difference between the two cited formulas.
- Dimension checks help identify whether covariance expressions can be combined.
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Full text
# How to perform Shanken (1992) correction for errors-in-variables issue?
# How to perform Shanken (1992) correction for errors-in-variables issue?
I have two questions pertaining to the Shanken correction:
- The formula of Shanken correction shown in the Cochrane (2001) Asset Pricing book is as follow:
$$\sigma^2(\hat{\lambda}_{OLS})=1/T[(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}(1+\lambda^{'}\Sigma_{f}^{-1}\lambda)+\Sigma_{f}]$$
I think I did not understand the formula correctly as I think the multiplicative term will result in a scalar, whereas the additive term will be in matrix form given that $\Sigma_{f}$ is the variance-covariance matrix of factors. So, it's impossible to add a scalar and a matrix, right? So, I might misunderstand it. I have looked through some lecture examples online, most of them dealing with a single factor (i.e. CAPM beta), hence the $\Sigma_{f}$ is simply the variance of the market excess returns. But I'm wondering how am I going to compute this correction if I have multiple factors (e.g. Fama-French three-factor model)? Do I need to compute the variance-covariance matrix of all factors or only employ the variance of a relevant factor in calculating the adjusted standard error?
- The formula stated in Shanken (1992) also seemed to be slightly different to me:
$$(1+c)[\hat{W}-\hat{\Sigma}_{F}]+\hat{\Sigma_{F}}$$
I'm wondering why this formula has an additional term, $\hat{\Sigma}_{F}$, to be subtracted from the sample covariance matrix, $\hat{W}$, as compared to the formula above.
## Answer by Richard Hardy (score 1)
https://quant.stackexchange.com/a/77021
### Question 1
If there are $k=1$ factors (i.e. a single factor):
- $\beta$ is a vector (a single-column matrix),
- $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}$ is a scalar,
- $\lambda^{'}\Sigma_{f}^{-1}\lambda$ is a scalar and thus
- $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}(1+\lambda^{'}\Sigma_{f}^{-1}\lambda)$ is a scalar, matching
- $\Sigma_{f}$ that is a scalar.
If there are $k>1$ factors:
- $\beta$ is a $k$-column matrix,
- $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}$ is a matrix
- $\lambda^{'}\Sigma_{f}^{-1}\lambda$ is a scalar and thus
- $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}(1+\lambda^{'}\Sigma_{f}^{-1}\lambda)$ is a matrix, matching
- $\Sigma_{f}$ that is a matrix.
The dimensions seem to match in both cases.
### Question 2
See the summary of Shanken (1992) laid out nicely in this answer on Cross Validated. It seems Cochrane's (2005) treatment omits some details, and the equation you give is not considered explicitly by Cochrane.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.