Covariance Eigenvalue Filtering and Shrinkage Methods
Summary
The note explains why a sample covariance matrix may have unreliable or zero eigenvalues and summarizes ways to address the problem. Even with balanced observations drawn independently from one distribution, sample eigenvalues differ from population values. When the number of observations is smaller than the number of instruments, the sample covariance matrix is singular, with zero eigenvalues, and cannot be inverted. This motivates modifying or estimating the covariance matrix before using it in portfolio or risk calculations.
Methods named in the response include Ledoit–Wolf shrinkage, random matrix theory, sparse rotations, graphical lasso, and spiked covariance models. A separate answer describes eigendecomposing the matrix and replacing near-zero eigenvalues with a small positive value, while suggesting shrinkage as a better approach. The discussion does not specify which method Meucci used in the cited case study, nor compare methods empirically. It offers a menu of approaches rather than implementation guidance or evidence that one choice is best in all settings.
Key ideas
- Sample covariance eigenvalues can differ from the corresponding population eigenvalues.
- When the observation count is below the instrument count, a sample covariance matrix can be singular.
- Ledoit–Wolf shrinkage, random matrix methods, sparse rotations, and graphical lasso are listed as possible remedies.
- Replacing near-zero eigenvalues with a small positive value is presented as a crude repair.
- The discussion does not identify the specific filtering method used in the cited case study.
Tags
Full text
# Filtering smallest eigenvalues
# Filtering smallest eigenvalues
In Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors [1], which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on pages 9-10, Section 6, Case study: security-based investment.
> [...] We estimate every month the covariance matrix of the of the [sic] S&P stock returns $\Sigma_{F}$ using a one-year rolling window of daily observations, and filtering the smallest eigenvalues to ensure positive definiteness.
What is meant by "filtering the smallest eigenvalues"? What methods are there to transform the covariance matrix in such a way that the smallest eigenvalues (and corresponding eigenvectors) get removed, but the rest stay intact, and which one is used here in particular?
Edit: This SE question [2] is highly relevant to this one.
[1] https://papers.ssrn.com/sol3/Papers.cfm?abstract_id=2276632
[2] What is the best way to "fix" a covariance matrix that is not positive semi-definite?
## Answer by Hans-Peter Schrei (score 3, accepted)
https://quant.stackexchange.com/a/40728
After writing an email to Meucci directly, I posted the question in his LinkedIn Group ARPM - Advanced Risk and Portfolio Management. Below are his answer and the answer of other group members, which echo the answer and comments already given here on SE.
Attilio Meucci Suppose that you are in an ideal world where
- you have a perfectly balanced panel of data n_ (number of instruments) x t_ (number of observations)
- each n_ dimensional column is a realization from the same joint distribution
- all such realizations are independent across time Even in the above ideal world, the sample covariance eigenvalues will be different from the true (population) covariance eigenvalues (https://www.arpm.co/lab/redirect.php?permalink=exam-ydaosw-copy-19) In particular, in the extreme case where t_ < n_ the sample covariance is not invertible, which means that the lowest eigenvalue(s) are zero.
To "fix" the above issue, there are a variety of techniques
- Ledoit-Wolf (https://www.arpm.co/lab/redirect.php?permalink=covariance-shrinkage-ledoit-wolf)
- random matrix theory (https://www.arpm.co/lab/redirect.php?permalink=correlation-shrinkage-random-matrix-theory)
- sparse rotations (https://www.arpm.co/lab/redirect.php?permalink=covariance-shrinkage-spars)
- GLASSO (https://www.arpm.co/lab/redirect.php?permalink=correlation-shrinkage-markov-network) etc...
- Spiked Covariance Model https://arxiv.org/abs/1311.0851 (contributed by Marc Weibel)
- Simonian, 2010: Simulation based on eigen-decomposed matrix (contributed by J.D. Opdyke) https://www.researchgate.net/publication/227601407_The_most_simple_methodology_to_create_a_valid_correlation_matrix_for_risk_management_and_option_pricing_purposes
## Answer by userid is i (score 0)
https://quant.stackexchange.com/a/40599
The covariance matrices must be nonnegative definite, so the smallest eigenvalues would be 0 or slightly positive. A crude method would be to take the orthogonal decomposition UDU' and shift the near- zero entries in D to epsilon. Shrinkage estimation of covariance seems 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.