Choosing Between Covariance and Correlation Matrix Shrinkage

Summary

The document compares shrinking a sample covariance matrix with shrinking a sample correlation matrix before using the result to forecast variance. Correlation shrinkage can be transformed back using sample variances, while direct covariance shrinkage also adjusts variance estimates toward a common target. The discussion favors covariance shrinkage when asset variances differ substantially, because correlation-based shrinkage can treat those assets too similarly and discard useful scale information.

The evidence is a practical observation that covariance shrinkage produced more accurate forward variance forecasts for the questioner. No dataset, shrinkage target, estimation procedure, or out-of-sample comparison is provided, so this is a contextual recommendation rather than a general empirical result. The better choice may depend on the assets, covariance structure, and forecasting objective.

Key ideas

• Shrinking covariance directly also affects estimates of individual variances.
• Shrinking correlation and restoring sample variances retains those original variance estimates.
• Correlation shrinkage may underuse scale differences when asset variances vary substantially.
• The stated preference for covariance shrinkage rests on an informal forecasting experience.

Tags

# Shrink covariance or correlation matrix


# Shrink covariance or correlation matrix












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

## Answer by Python31241 (score 2)

https://quant.stackexchange.com/a/55581

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.

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.