# Cleansing covariance matrices via Random matrix theory

I am exploring de-noising and cleansing of covariance matrices via Random Matrix Theory. RMT is a competitor to shrinkage methods of covariance estimation. There are various methods expressed usually by the names of the authors: LPCB, PG+, and so on.

For each method, one can start by filtering the covariance matrix directly, or filter the correlation matrix and then covert the cleansed correlation matrix into a covariance matrix. My question involves the latter case.

I have noticed that when cleaning a correlation matrix that the resulting diagonal is not a diagonal of 1s (as one would expect to see in a correlation matrix).

My question -- when constructing a cleansed covariance matrix by first filtering its corresponding correlation matrix, does one:

1. "Fix" the diagonals of the intermediate cleansed correlation matrix to a diagonal of 1s before finally converting it back to a covariance matrix? This seems to be the case with the PG+ method, but not the LPCB method.

2. Or does one convert the cleansed correlation matrix to a covariance matrix and then fix the diagonal of the resulting covariance matrix to the diagonal of the original covariance matrix?

In both cases the trace is preserved.

• I think RMT only helps you filter out noise eigenvectors. How we re-construct the correlation matrix after that is totally up to us. I personally will choose an approach that yield a diagonal of 1's. As for a diagonal != 1, it feels like that they are computing the 'cross-'correlation matrix between clean (after RMT) and noisy (before) time series (cause correlation of identical time series must equal to 1). Is it what you want? I thought we are looking for a correlation matrix composed of clean time series only. – 楊祝昇 Oct 24 '11 at 17:37
• Correct - RMT will identify the upper noise band or cut-off value for eigenvalues. I suppose since that is all we can ask of RMT then after this it is just up to us on how to proceed. Yes, we are looking for a correlation matrix of clean time-series only. The issue arises because when you de-noise the eigenvalues there are invariably changes to the diagonal of the correlation matrix from this process. I will test out both procedures and post my findings in a couple of days... – Ram Ahluwalia Oct 26 '11 at 3:05

I tested both procedures. The results are virtually indistinguishable - the decision is not consequential. I opted for approach #1.

This is a very good question. In part, you can find a comparison by going to randommatrixportfolios.com and looking at the wealth charts for e.g. the Dow 30 portfolios, say, the 2-year data. You will note that portfolios based regressing the log-returns of price on the "signal" PCs (principal components) based on the Marcenko-Pastur noise cutoff and using the residuals as price returns in the portfolio resulted in much more wealth after two years when compared with (i) removing the main market component as well as the 3 shrinkage methods. The various portfolios were :

-MinVar - Minimum variance portfolio, tangency is used for unbalanced, whereas minimum variance is use for rebalanced portfolios.

-EWMA - Exponential weighted moving average determination of returns and standard deviation.

-RES - Component subtraction used to remove effect of the first principal component of the correlation matrix on returns.

-MP - Component subtraction used to remove effect of noise eigenvectors (below Marcenko-Pastur cutoff, lambda+), on returns.

-RESMP - Component subtraction employed to remove effects of greatest principal component and noise eigenvectors below MP cutoff.

-DK - Daniels-Kass shrinkage of correlation matrix.

-LW - Ledoit-Wolf shrinkage of correlation matrix.

-SS - Schafer-Strimmer shrinkage of correlation matrix.

There are portfolios, however, for which the shrinkage methods result in more wealth, but overall we like the MP (Marcenko-Pastur noise-signal) component removal method.

We also removed volatility clustering from the raw log-returns of price data using ARCH(1)/GARCH(1,1) models, and the results were not that different.

As a totally relevant aside, we wrote an entire chapter on covariance filtering methods, introduced conjectures for MP, and equations which could be used to develop algorithms in Chap 28 of enter link description here.