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Shrinkage and Positive Definiteness in Newey–West Covariance Estimation

Article Quant Q&A · Author: Richi Wa
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Summary

The document raises a portfolio optimization question about shrinkage estimation of a Newey–West covariance matrix. It describes combining the contemporaneous covariance matrix with a symmetrized lagged covariance term, and considers using an identity matrix as the shrinkage target for the contemporaneous component. It asks whether the zero matrix is an appropriate prior for the lagged component and whether shrinkage methods are available for lag covariance matrices.

The central issue is that lagged covariance matrices need not be positive definite, while the resulting covariance estimate may lose positive definiteness if the contemporaneous and lagged terms are estimated or regularized separately. The note cites the Newey–West construction as a positive semidefinite heteroskedasticity and autocorrelation consistent estimator, but provides no proposed shrinkage solution or implementation. Its value is in identifying the constraint that a portfolio optimization workflow must address.

Key ideas

  • The described Newey–West estimate combines contemporaneous covariance with symmetrized lagged covariance.
  • An identity matrix is proposed as a shrinkage target for contemporaneous covariance.
  • Lagged covariance matrices need not be positive definite, complicating direct shrinkage.
  • Separately shrinking components may fail to preserve positive definiteness of the final estimate.
  • The document poses the estimation problem but does not provide a solution or software recommendation.

Tags

StatisticsPortfolio constructionRisk management
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# Shrinkage Estimator for Newey-West Covariance Matrix


# Shrinkage Estimator for Newey-West Covariance Matrix












I like to apply the Newey-West covariance estimator for portfolio optmization which is given by $$ \Sigma = \Sigma(0) + \frac12 \left (\Sigma(1) + \Sigma(1)^T \right), $$ where $\Sigma(i)$ is the lag $i$ covariance matrix for $i=0,1$. Furthermore I like to use shrinkage estimators as implemented in the corpcor package for R. The identity matrix as shrinkage prior for $\Sigma(0)$ is plausible.

What would you use as prior for $\Sigma(1)$ - the zero-matrix? Do you know an R implementation that allows to estimate lag-covariance matrices using shrinkage? There must be some basic difference as a lag-covariance matrix is not necessarily positive-definite (e.g. the zero-matrix). If I apply shrinkage to $\Sigma(0)$ and use the standard sample-estimator for $\Sigma(1)$ then it is not assured that $\Sigma$ is positive-definite.

EDIT: The above definition is taken from:

Whitney K. Newey and Keneth D. West. A simple, positive semi-denite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3):703-708, 1987.

It can also be found here in formula (1.9) on page 6.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.