# Shrinkage Estimator for Newey-West Covariance Matrix

This is a cross post. I would like to apply the Newey-West covariance estimator for portfolio optmization. Up to lag one it 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.

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.