CovEst.2003LW {CovTools}

Covariance Estimation with Linear Shrinkage

Description

Ledoit and Wolf (2003, 2004) proposed a linear shrinkage strategy to estimate covariance matrix with an application to portfolio optimization. An optimal covariance is written as a convex combination as follows,

\hat{\Sigma} = \delta \hat{F} + (1-\delta) \hat{S}

where \delta \in (0,1) a control parameter/weight, \hat{S} an empirical covariance matrix, and \hat{F} a target matrix. Although authors used F a highly structured estimator, we also enabled an arbitrary target matrix to be used as long as it's symmetric and positive definite of corresponding size.

Usage

CovEst.2003LW(X, target = NULL)

Arguments

X

an (n\times p) matrix where each row is an observation.

target

target matrix F. If target=NULL, constant correlation model estimator is used. If target is specified as a qualified matrix, it is used instead.

Value

a named list containing:

S

a (p\times p) covariance matrix estimate.

delta

an estimate for convex combination weight according to the relevant theory.

References

Ledoit O, Wolf M (2003). “Improved estimation of the covariance matrix of stock returns with an application to portfolio selection.” Journal of Empirical Finance, 10(5), 603–621. ISSN 09275398.

Ledoit O, Wolf M (2004). “A well-conditioned estimator for large-dimensional covariance matrices.” Journal of Multivariate Analysis, 88(2), 365–411. ISSN 0047259X.

Ledoit O, Wolf M (2004). “Honey, I Shrunk the Sample Covariance Matrix.” The Journal of Portfolio Management, 30(4), 110–119. ISSN 0095-4918.

Examples

## CRAN-purpose small computation
# set a seed for reproducibility
set.seed(11)

#  small data with identity covariance
pdim      <- 5
dat.small <- matrix(rnorm(20*pdim), ncol=pdim)

#  run the code with highly structured estimator
out.small <- CovEst.2003LW(dat.small)

#  visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(diag(5)[,pdim:1], main="true cov")
image(cov(dat.small)[,pdim:1], main="sample cov")
image(out.small$S[,pdim:1], main="estimated cov")
par(opar)

## Not run: 
## want to see how delta is determined according to
#  the number of observations we have.
nsamples = seq(from=5, to=200, by=5)
nnsample = length(nsamples)

#  we will record two values; delta and norm difference
vec.delta = rep(0, nnsample)
vec.normd = rep(0, nnsample)
for (i in 1:nnsample){
  dat.norun <- matrix(rnorm(nsamples[i]*pdim), ncol=pdim) # sample in R^5
  out.norun <- CovEst.2003LW(dat.norun)                   # run with default

  vec.delta[i] = out.norun$delta
  vec.normd[i] = norm(out.norun$S - diag(pdim),"f")       # Frobenius norm
}

# let's visualize the results
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(nsamples, vec.delta, lwd=2, type="b", col="red", main="estimated deltas")
plot(nsamples, vec.normd, lwd=2, type="b", col="blue",main="Frobenius error")
par(opar)

## End(Not run)