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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?

  1. 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 is this formula have an additional term, $\hat{\Sigma}_{F}$, to be subtracted from the sample covariance matrix, $\hat{W}$, as compared to the formula bove.

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