rOWCov: Realized Outlyingness Weighted Covariance
In highfrequency: Tools for Highfrequency Data Analysis

Description Usage Arguments Details Value Author(s) References Examples

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{∑_{i=1}^{M}w(d_{t,i})r_{t,i}r'_{t,i}}{\frac{1}{M}∑_{i=1}^{M}w(d_{t,i})},

The weight w_{i,Δ} is one if the multivariate jump test statistic for r_{i,Δ} 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.

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rOWCov(rdata, cor = FALSE, align.by = NULL, align.period = NULL, 
       makeReturns = FALSE, seasadjR = NULL, wfunction = "HR", alphaMCD = 0.75, 
       alpha = 0.001, ...)

`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.
`align.by`	a string, align the tick data to "seconds"\|"minutes"\|"hours".
`align.period`	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).
`...`	additional arguments.

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

an N x N matrix

Jonathan Cornelissen and Kris Boudt

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

 # Realized Outlyingness Weighted Variance/Covariance for CTS aligned   
 # at 5 minutes.
 data(sample_tdata); 
 data(sample_5minprices_jumps);
 
 # Univariate: 
 rvoutw = rOWCov( rdata = sample_tdata$PRICE, align.by ="minutes", 
                    align.period =5, makeReturns=TRUE); 
 rvoutw 
 
 # Multivariate: 
 rcoutw = rOWCov( rdata = sample_5minprices_jumps['2010-01-04'], makeReturns=TRUE); 
 rcoutw