Is it preferable to shrink the covariance matrix vs the correlation matrix? Technically this amounts to either shrinking the sample correlation matrix and then transforming the shrunk correlation matrix using the sample variances versus just shrinking the sample covariance matrix all in one go (this has the effect of shrinking the variances to tr(A)/n)

FWIW, I’ve found in practice that shrinking the covariance matrix leads to more accurate forecasts of forward variance


Generally it is better to shrink the covariance matrix—since the variances of your data probably vary a lot, and the correlation matrix treats them all as essentially equal variance, you throw out the baby with the bath water by pausing to the correlation matrix. In effect, when you shrink the correlation matrix, you correct a lot of stuff that is not important. So it is not surprising at all that you find shrinking the cov matrix to work better.

  • $\begingroup$ thanks! what do you mean by "correct a lot of stuff that is not important" $\endgroup$ – Michael Jul 16 '20 at 17:28
  • $\begingroup$ I mean, the correction will be heavily influenced by factors that appear important in the correlation matrix but are not important in the covariance matrix, which is probably what matters to you. Ie, you probably care about risk. Almost certainly the correlation matrix is not what really matters—depending on the application, if you are lucky, you might be able to characterize what matters more precisely and condition on that. $\endgroup$ – Python31241 Jul 28 '20 at 16:56

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