CovEst.2003LW {CovTools} R Documentation

## 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{Σ} = δ \hat{F} + (1-δ) \hat{S}

where δ \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
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