itheta_vcov {MarkowitzR}

Compute variance covariance of Inverse 'Unified' Second Moment

Description

Computes the variance covariance matrix of the inverse unified second moment matrix.

Usage

itheta_vcov(X,vcov.func=vcov,fit.intercept=TRUE)

Arguments

X

an n \times p matrix of observed returns.

vcov.func

a function which takes an object of class lm, and computes a variance-covariance matrix. If equal to the string "normal", we assume multivariate normal returns.

fit.intercept

a boolean controlling whether we add a column of ones to the data, or fit the raw uncentered second moment. For now, must be true when assuming normal returns.

Details

Given p-vector x with mean \mu and covariance, \Sigma, let y be x with a one prepended. Then let \Theta = E\left(y y^{\top}\right), the uncentered second moment matrix. The inverse of \Theta contains the (negative) Markowitz portfolio and the precision matrix.

Given n contemporaneous observations of p-vectors, stacked as rows in the n \times p matrix X, this function estimates the mean and the asymptotic variance-covariance matrix of \Theta^{-1}.

One may use the default method for computing covariance, via the vcov function, or via a 'fancy' estimator, like sandwich:vcovHAC, sandwich:vcovHC, etc.

Value

a list containing the following components:

mu

a q = (p+1)(p+2)/2 vector of 1 + squared maximum Sharpe, the negative Markowitz portfolio, then the vech'd precision matrix of the sample data

Ohat

the q \times q estimated variance covariance matrix.

n

the number of rows in X.

pp

the number of assets plus as.numeric(fit.intercept).

Note

By flipping the sign of X, the inverse of \Theta contains the positive Markowitz portfolio and the precision matrix on X. Performing this transform before passing the data to this function should be considered idiomatic.

A more general form of this function exists as mp_vcov.

Replaces similar functionality from SharpeR package, but with modified API.

Author(s)

Steven E. Pav [e-mail]

References

Pav, S. E. "Asymptotic Distribution of the Markowitz Portfolio." 2013 https://arxiv.org/abs/1312.0557

Pav, S. E. "Portfolio Inference with this One Weird Trick." R in Finance, 2014 http://past.rinfinance.com/agenda/2014/talk/StevenPav.pdf

See Also

theta_vcov, mp_vcov.

Examples

X <- matrix(rnorm(1000*3),ncol=3)
# putting in -X is idiomatic:
ism <- itheta_vcov(-X)
iSigmas.n <- itheta_vcov(-X,vcov.func="normal")
iSigmas.n <- itheta_vcov(-X,fit.intercept=FALSE)
# compute the marginal Wald test statistics:
qidx <- 2:ism$pp
wald.stats <- ism$mu[qidx] / sqrt(diag(ism$Ohat[qidx,qidx]))

# make it fat tailed:
X <- matrix(rt(1000*3,df=5),ncol=3)
ism <- itheta_vcov(X)
qidx <- 2:ism$pp
wald.stats <- ism$mu[qidx] / sqrt(diag(ism$Ohat[qidx,qidx]))

if (require(sandwich)) {
 ism <- itheta_vcov(X,vcov.func=vcovHC)
 qidx <- 2:ism$pp
 wald.stats <- ism$mu[qidx] / sqrt(diag(ism$Ohat[qidx,qidx]))
}

# add some autocorrelation to X
Xf <- filter(X,c(0.2),"recursive")
colnames(Xf) <- colnames(X)
ism <- itheta_vcov(Xf)
qidx <- 2:ism$pp
wald.stats <- ism$mu[qidx] / sqrt(diag(ism$Ohat[qidx,qidx]))

if (require(sandwich)) {
ism <- itheta_vcov(Xf,vcov.func=vcovHAC)
 qidx <- 2:ism$pp
 wald.stats <- ism$mu[qidx] / sqrt(diag(ism$Ohat[qidx,qidx]))
}