How to perform Shanken (1992) correction for errors-in-variables issue?

Question

I have two questions pertaining to the Shanken correction:

1. 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?

2. 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 1

Question 1

If there are $k=1$ factors (i.e. a single factor):

1. $\beta$ is a vector (a single-column matrix),
2. $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}$ is a scalar,
3. $\lambda^{'}\Sigma_{f}^{-1}\lambda$ is a scalar and thus
4. $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}(1+\lambda^{'}\Sigma_{f}^{-1}\lambda)$ is a scalar, matching
5. $\Sigma_{f}$ that is a scalar.

If there are $k>1$ factors:

1. $\beta$ is a $k$-column matrix,
2. $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}$ is a matrix
3. $\lambda^{'}\Sigma_{f}^{-1}\lambda$ is a scalar and thus
4. $(\beta^{'}\beta)^{-1}\beta^{'}\Sigma\beta(\beta^{'}\beta)^{-1}(1+\lambda^{'}\Sigma_{f}^{-1}\lambda)$ is a matrix, matching
5. $\Sigma_{f}$ 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.