Interpreting Canonical Correlations Between PCA and Pricing Factors

Summary

The document presents questions about comparing principal components extracted from a global excess-return covariance matrix with a set of asset pricing factors using canonical correlation analysis. It reports five canonical correlations, with the leading value near one and the remaining values also relatively high. The author asks how to interpret these paired correlations and what they imply about the relationship between the two sets of variables.

It also asks what is meant by eigenvector weights for principal components, and whether those weights are the square roots of eigenvalues. The text contains no answer, derivation, or empirical assessment beyond the reported output, so it does not establish how the canonical pairs should be interpreted or resolve the distinction between eigenvectors, loadings, and eigenvalues. It is useful as a prompt for studying multivariate factor comparison and PCA terminology, but its conclusions are limited to the questions posed.

Key ideas

• Canonical correlation analysis compares linear combinations of two variable sets.
• The example compares principal components of excess returns with asset pricing factors.
• The reported output contains five canonical correlations.
• The author asks how to interpret the strength of the canonical relationships.
• The document does not resolve whether principal component weights are related to square-root eigenvalues.

Tags

# Some questions to canonical correlations between principle components and asset pricing factors using R


# Some questions to canonical correlations between principle components and asset pricing factors using R












I have done a asympotical principle component analysis (APCA), using `eigen()` in R, of the covariance matrix of a global dataset of excess returns. I took the resulting eigenvectors (E) of the top principle components as a set and compared them with another set containing 5 asset pricing factors (F) using the function `cancor(F,E)`, ending in 5 canonical pairs with canonical correlations

```
    $cor
[1] 0.9538597 0.8230864 0.8126250 0.8076996 0.7953763
```

- Now the question arises for me, how exactly I have to understand this output and how I can interpret these numbers?

- Furthermore, I wonder what is generally meant by "Eigenvector weights for the principle components"? Are these $\sqrt{Eigenvalues}$ ?

Thank you in advance!

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