highfrequency (version 0.7.0.1)

rOWCov: Realized Outlyingness Weighted Covariance

Description

Function returns the Realized Outlyingness Weighted Covariance, defined in Boudt et al. (2008).

Let \(r_{t,i}\), for \(i=1,...,M\) be a sample of \(M\) high-frequency \((N x 1)\) return vectors and \(d_{t,i}\) their outlyingness given by the squared Mahalanobis distance between the return vector and zero in terms of the reweighted MCD covariance estimate based on these returns.

Then, the rOWCov is given by $$ \mbox{rOWCov}_{t}=c_{w}\frac{\sum_{i=1}^{M}w(d_{t,i})r_{t,i}r'_{t,i}}{\frac{1}{M}\sum_{i=1}^{M}w(d_{t,i})}, $$ The weight \(w_{i,\Delta}\) is one if the multivariate jump test statistic for \(r_{i,\Delta}\) in Boudt et al. (2008) is less than the 99.9% percentile of the chi-square distribution with \(N\) degrees of freedom and zero otherwise. The scalar \(c_{w}\) is a correction factor ensuring consistency of the rOWCov for the Integrated Covariance, under the Brownian Semimartingale with Finite Activity Jumps model.

Usage

rOWCov(
  rData,
  cor = FALSE,
  alignBy = NULL,
  alignPeriod = NULL,
  makeReturns = FALSE,
  seasadjR = NULL,
  wFunction = "HR",
  alphaMCD = 0.75,
  alpha = 0.001
)

Arguments

rData

a \((M x N)\) matrix/zoo/xts object containing the \(N\) return series over period \(t\), with \(M\) observations during \(t\).

cor

boolean, in case it is TRUE, the correlation is returned. FALSE by default.

alignBy

a string, align the tick data to "seconds"|"minutes"|"hours".

alignPeriod

an integer, align the tick data to this many [seconds|minutes|hours].

makeReturns

boolean, should be TRUE when rData contains prices instead of returns. FALSE by default.

seasadjR

a \((M x N)\) matrix/zoo/xts object containing the seasonaly adjusted returns. This is an optional argument.

wFunction

determines whether a zero-one weight function (one if no jump is detected based on \(d_{t,i}\) and 0 otherwise) or Soft Rejection ("SR") weight function is to be used. By default a zero-one weight function (wFunction = "HR") is used.

alphaMCD

a numeric parameter, controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75. See Boudt et al. (2008) or the covMcd function in the robustbase package.

alpha

is a parameter between 0 en 0.5, that determines the rejection threshold value (see Boudt et al. (2008) for details).

Value

an \(N x N\) matrix

Details

Advantages of the rOWCov compared to the rBPCov include a higher statistical efficiency, positive semidefiniteness and affine equivariance. However, the rOWCov suffers from a curse of dimensionality. Indeed, the rOWCov gives a zero weight to a return vector if at least one of the components is affected by a jump. In the case of independent jump occurrences, the average proportion of observations with at least one component being affected by jumps increases fast with the dimension of the series. This means that a potentially large proportion of the returns receives a zero weight, due to which the rOWCov can have a low finite sample efficiency in higher dimensions

References

Boudt, K., C. Croux, and S. Laurent (2008). Outlyingness weighted covariation. Mimeo.

Examples

# NOT RUN {
# Realized Outlyingness Weighted Variance/Covariance for prices aligned   
# at 5 minutes.

# Univariate: 
rvoutw <- rOWCov(rData = sampleTData$PRICE, alignBy = "minutes",
                   alignPeriod = 5, makeReturns = TRUE)
rvoutw 

# Multivariate: 
rcoutw <- rOWCov(rData = sample5MinPricesJumps['2010-01-04'], makeReturns = TRUE)
rcoutw

# }