# 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.

• I just rearead the comments here which say that the shrinkage prior is the zero-matrix for correlations ... this makes things somewhat better. – Ric Jul 24 '13 at 13:45
• My understanding of Newey-West estimators is that it is used to calculate the covariance matrix of parameters in a regression. I could imagine using it in a robust portfolio optimization (concerned with uncertainty in the mean parameters), but whether it makes sense to use it for estimating the covariance of returns, I don't know. That being said, why are you not able to simply replace the sample covariance in the shrinkage formula with your new Newey-West estimate? – John Jul 25 '13 at 15:10
• @John I know that NW is usually used in the context of regressions errors. There are situations where your assets in portfolio optimization show auto-correlation (daily returns, US and Japan e.g.) - then just using the ordinary covariance matrix will be misleading. We have published this preprint in JOR about consequences of this issue. Now I want to combine shrinkage with it (not for a paper but for practical purposes). – Ric Jul 25 '13 at 15:42
• You may want to add the link to the preprint in the question. I still am curious as to what is the issue with simply replacing the sample estimate with the Newey West estimate. Anyway, the original paper implies that it is estimated through GMM rather than Maximum Likelihood. I would have suggested finding the Bayesian prior that would be equivalent to whatever shrinkage you want to make and then adopting the approach for the Newey-West estimator, but since it is not based on ML I'm not sure if that would work. – John Jul 25 '13 at 17:31