Multi Anchor Point Shrinkage for the Sample Covariance Matrix (Extended Version)
Hubeyb Gurdogan, Alec Kercheval
Portfolio managers faced with limited sample sizes must use factor models to estimate the covariance matrix of a high-dimensional returns vector. For the simplest one-factor market model, success rests on the quality of the estimated leading eigenvector "beta".
When only the returns themselves are observed, the practitioner has
available the "PCA" estimate equal to the leading eigenvector of the
sample covariance matrix. This estimator performs poorly in various
ways. To address this problem in the high-dimension, limited sample
size asymptotic regime and in the context of estimating the minimum
variance portfolio, Goldberg, Papanicolau, and Shkolnik (Goldberg et
al. [2021]) developed a shrinkage method (the "GPS estimator") that
improves the PCA estimator of beta by shrinking it toward the target
unit vector with constant entries.
In this paper we continue their work to develop a more general
framework of shrinkage targets that allows the practitioner to make
use of further information to improve the estimator. Examples include
sector separation of stock betas, and recent information from prior
estimates. We prove some precise statements and illustrate the
resulting improvements over the GPS estimator with some numerical
experiments.