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I'm trying to estimate the correlation matrix of 50 stock returns in R. I'm aware that i should shrink the correlation matrix, but I'm also interested in using the Spearman’s rank correlation since it dosen't require the distribution to be normal.

For the Sperman correlation I usually specify the method = "spearman" in the cov() function, which dosen't allow shrinkage tecniques. For shrinkage I've been looking at the tawny package and the corpcor package, but they're not allowing for any specification of the method pearson/spearman etc.

Any suggestions on how I could handle this issue?

eaoestergaard
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    Have you looked at the package **ShrinkCovMat**? From the description: *"Provides nonparametric Stein-type shrinkage estimators of the covariance matrix that are suitable and statistically efficient when the number of variables is larger than the sample size. These estimators are non-singular and well-conditioned regardless of the dimensionality."* – KoenV Apr 03 '17 at 18:44
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    Yes, and corpcor + tawn too. However, when estimating the correlationmatrix they're probably not using the Spearman correlation. – eaoestergaard Apr 04 '17 at 13:54
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    I am sorry but I don't think I can help you any further. I have upvoted your question, hoping someone else can help you. The only link that may be of interest to you that I have found, is the following one. It points to a bachelor thesis handling shrinkage and deals with Spearman: [link](http://pages.pomona.edu/~jsh04747/Student%20Theses/BrianWilliamson14.pdf) . If this article is of interest to you, you could consider contacting the student or the staff member of Pomona College. – KoenV Apr 04 '17 at 15:22
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    Thank you, I'll take a look! – eaoestergaard Apr 05 '17 at 12:07
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    I flagged this question to the moderator, as I believe it is a good question which received too little attention. Let's see what happens. Good Luck again. – KoenV Apr 06 '17 at 10:38

1 Answers1

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If you are interested in using a non-parametric correlation, perhaps you can look at Kendall's tau. As a U-statistics, it is asymptotically normal. (If I am not mistaken, Spearman too is asymptotically normal, so the procedure I am about to describe is valid for Spearman as well) Hence, it suggests you can use the normal log-likelihood to perform your shrinkage.

More precisely, let tau.hat be your vectorized (estimated) correlation matrix and

L(tau | tau.hat, Sigma) = t(tau - tau.hat) Sigma^{-1} (tau - tau.hat)

where cov(tau.hat) = Sigma. L is the loss function that will allow you to know when you applied too much shrinkage (realizing that L should be roughly chi squared distributed).

This is not very helpful since you don't know Sigma, but there are ways of estimating it (for 50 stocks it might still be okay, but that explodes quickly). This is the main reason why I suggest using Kendall's tau over Spearman's rho: you can have an idea of its variance (Sigma). (See this paper for an estimator of Sigma: https://arxiv.org/abs/1706.05940)

From there, you can use a standard technique to shrink Sigma.hat (this is necessary), e.g. simply shrink it towards its diagonal version Sigma0.hat with

Sigma.tilde = w Sigma0.hat + (1-w) Sigma.hat

with, say, w = .5. At least make sure it is positive definite...

Then (finally!) get a shrunk version of tau.hat by letting q goes to 1 in tau.tilde = (1-q) tau.hat + q mean(tau.hat) until Prob[chi_d^2 > L(tau.tilde | tau.hat, Sigma.tilde)] = .05, where the degrees of freedom for you chi squared are (?!?!?) d = length(tau.hat) - q.

I am not sure the degrees of freedom for the chi squared are right. Actually, I am pretty sure they are not. Note also that .05 was chosen somewhat arbitrarily. Bottom line is that you might want to take a look at the paper referred to above, as they (we, I must say) indeed shrink the Kendall correlation matrix of log-returns from 100 stocks. The way it is done allows us to know more about the degrees of freedom in question (and does not require you to provide a structure a priori as it learns a block structure from the data).

Perochkin
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