rRTSCov: Robust two time scale covariance estimation

In jonathancornelissen/highfrequency: Tools for Highfrequency Data Analysis

Robust two time scale covariance estimation

Description

Calculate the robust two time scale covariance matrix proposed in Boudt and Zhang (2010). Unlike the rOWCov, but similarly to the rThresholdCov, the rRTSCov uses univariate jump detection rules to truncate the effect of jumps on the covariance estimate. By the use of two time scales, this covariance estimate is not only robust to price jumps, but also to microstructure noise and non-synchronic trading.

Usage

rRTSCov(
  pData,
  cor = FALSE,
  startIV = NULL,
  noisevar = NULL,
  K = 300,
  J = 1,
  KCov = NULL,
  JCov = NULL,
  KVar = NULL,
  JVar = NULL,
  eta = 9,
  makePsd = FALSE,
  ...
)

Arguments

pData	a list. Each list-item i contains an xts object with the intraday price data of stock i for day t.
cor	boolean, in case it is TRUE, and the input data is multivariate, the correlation is returned instead of the covariance matrix. FALSE by default.
startIV	vector containing the first step estimates of the integrated variance of the assets, needed in the truncation. Is NULL by default.
noisevar	vector containing the estimates of the noise variance of the assets, needed in the truncation. Is NULL by default.
K	positive integer, slow time scale returns are computed on prices that are K steps apart.
J	positive integer, fast time scale returns are computed on prices that are J steps apart.
KCov	positive integer, for the extradiagonal covariance elements the slow time scale returns are computed on prices that are K steps apart.
JCov	positive integer, for the extradiagonal covariance elements the fast time scale returns are computed on prices that are J steps apart.
KVar	vector of positive integers, for the diagonal variance elements the slow time scale returns are computed on prices that are K steps apart.
JVar	vector of positive integers, for the diagonal variance elements the fast time scale returns are computed on prices that are J steps apart.
eta	positive real number, squared standardized high-frequency returns that exceed eta are detected as jumps.
makePsd	boolean, in case it is TRUE, the positive definite version of rRTSCov is returned. FALSE by default.
...	used internally, do not change.

Details

The rRTSCov requires the tick-by-tick transaction prices. (Co)variances are then computed using log-returns calculated on a rolling basis on stock prices that are K (slow time scale) and J (fast time scale) steps apart.

The diagonal elements of the rRTSCov matrix are the variances, computed for log-price series X with n price observations at times τ_1,τ_2,…,τ_n as follows:

(1-\frac{\overline{n}_K}{\overline{n}_J})^{-1}(\{X,X\}_T^{(K)^{*}}-\frac{\overline{n}_K}{\overline{n}_J}\{X,X\}_T^{(J)^{*}}),

where \overline{n}_K=(n-K+1)/K, \overline{n}_J=(n-J+1)/J and

\{X,X\}_T^{(K)^{*}} =\frac{c_η^{*}}{K}\frac{∑_{i=1}^{n-K+1}(X_{t_{i+K}}-X_{t_i})^2I_X^K(i;η)}{\frac{1}{n-K+1}∑_{i=1}^{n-K+1}I_X^K(i;η)}.

The constant c_η adjusts for the bias due to the thresholding and I_{X}^K(i;η) is a jump indicator function that is one if

\frac{(X_{t_{i+K}}-X_{t_{i}})^2}{(\int_{t_{i}}^{t_{i+K}} σ^2_sds +2σ_{\varepsilon_{\mbox{\tiny X}}}^2)} \ \ ≤q \ \ η

and zero otherwise. The elements in the denominator are the integrated variance (estimated recursively) and noise variance (estimated by the method in Zhang et al, 2005).

The extradiagonal elements of the rRTSCov are the covariances. For their calculation, the data is first synchronized by the refresh time method proposed by Harris et al (1995). It uses the function refreshTime to collect first the so-called refresh times at which all assets have traded at least once since the last refresh time point. Suppose we have two log-price series: X and Y. Let Γ =\{ τ_1,τ_2,…,τ_{N^{\mbox{\tiny X}}_{\mbox{\tiny T}}}\} and Θ=\{θ_1,θ_2,…,θ_{N^{\mbox{\tiny Y}}_{\mbox{\tiny T}}}\} be the set of transaction times of these assets. The first refresh time corresponds to the first time at which both stocks have traded, i.e. φ_1=\max(τ_1,θ_1). The subsequent refresh time is defined as the first time when both stocks have again traded, i.e. φ_{j+1}=\max(τ_{N^{\mbox{\tiny{X}}}_{φ_j}+1},θ_{N^{\mbox{\tiny{Y}}}_{φ_j}+1}). The complete refresh time sample grid is Φ=\{φ_1,φ_2,...,φ_{M_N+1}\}, where M_N is the total number of paired returns. The sampling points of asset X and Y are defined to be t_i=\max\{τ\inΓ:τ≤q φ_i\} and s_i=\max\{θ\inΘ:θ≤q φ_i\}.

Given these refresh times, the covariance is computed as follows:

c_{N}( \{X,Y\}^{(K)}_T-\frac{\overline{n}_K}{\overline{n}_J}\{X,Y\}^{(J)}_T ),

where

\{X,Y\}^{(K)}_T =\frac{1}{K} \frac{∑_{i=1}^{M_N-K+1}c_i (X_{t_{i+K}}-X_{t_{i}})(Y_{s_{i+K}}-Y_{s_{i}})I_{X}^K(i;η) I_{Y}^K(i;η)}{\frac{1}{M_N-K+1}∑_{i=1}^{M_N-K+1}{I_X^K(i;η)I_Y^K(i;η)}},

with I_{X}^K(i;η) the same jump indicator function as for the variance and c_N a constant to adjust for the bias due to the thresholding.

Unfortunately, the rRTSCov is not always positive semidefinite. By setting the argument makePsd = TRUE, the function makePsd is used to return a positive semidefinite matrix. This function replaces the negative eigenvalues with zeroes.

Value

an N \times N matrix

Author(s)

Jonathan Cornelissen, Kris Boudt, and Emil Sjoerup.

References

Boudt K. and Zhang, J. 2010. Jump robust two time scale covariance estimation and realized volatility budgets. Mimeo.

Harris, F., McInish, T., Shoesmith, G., and Wood, R. (1995). Cointegration, error correction, and price discovery on informationally linked security markets. Journal of Financial and Quantitative Analysis, 30, 563-581.

Zhang, L., Mykland, P. A., and Ait-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 100, 1394-1411.

See Also

ICov for a list of implemented estimators of the integrated covariance.

Examples

## Not run: 
library(xts)
set.seed(123)
start <- strptime("1970-01-01", format = "%Y-%m-%d", tz = "UTC")
timestamps <- start + seq(34200, 57600, length.out = 23401)

dat <- cbind(rnorm(23401) * sqrt(1/23401), rnorm(23401) * sqrt(1/23401))

dat <- exp(cumsum(xts(dat, timestamps)))
price1 <- dat[,1]
price2 <- dat[,2]
rcRTS <- rRTSCov(pData = list(price1, price2))
# Note: List of prices as input
rcRTS

## End(Not run)

jonathancornelissen/highfrequency documentation built on Jan. 10, 2023, 7:29 p.m.