rRTSCov: Robust two time scale covariance estimation
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rRTSCov R Documentation
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 \tau_1,\tau_2,\ldots,\tau_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_\eta^{*}}{K}\frac{\sum_{i=1}^{n-K+1}(X_{t_{i+K}}-X_{t_i})^2I_X^K(i;\eta)}{\frac{1}{n-K+1}\sum_{i=1}^{n-K+1}I_X^K(i;\eta)}.
The constant c_\eta adjusts for the bias due to the thresholding and I_{X}^K(i;\eta) is a jump indicator function
that is one if
\frac{(X_{t_{i+K}}-X_{t_{i}})^2}{(\int_{t_{i}}^{t_{i+K}} \sigma^2_sds +2\sigma_{\varepsilon_{\mbox{\tiny X}}}^2)} \ \ \leq \ \ \eta
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 \Gamma =\{ \tau_1,\tau_2,\ldots,\tau_{N^{\mbox{\tiny X}}_{\mbox{\tiny T}}}\} and
\Theta=\{\theta_1,\theta_2,\ldots,\theta_{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.
\phi_1=\max(\tau_1,\theta_1). The subsequent refresh time is defined as the first time when both stocks have again traded, i.e.
\phi_{j+1}=\max(\tau_{N^{\mbox{\tiny{X}}}_{\phi_j}+1},\theta_{N^{\mbox{\tiny{Y}}}_{\phi_j}+1}). The
complete refresh time sample grid is
\Phi=\{\phi_1,\phi_2,...,\phi_{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\{\tau\in\Gamma:\tau\leq \phi_i\} and
s_i=\max\{\theta\in\Theta:\theta\leq \phi_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{\sum_{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;\eta)
I_{Y}^K(i;\eta)}{\frac{1}{M_N-K+1}\sum_{i=1}^{M_N-K+1}{I_X^K(i;\eta)I_Y^K(i;\eta)}},
with I_{X}^K(i;\eta) 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)
highfrequency documentation built on Oct. 4, 2023, 5:08 p.m.
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| rRTSCov | R Documentation |
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 |
cor |
boolean, in case it is |
startIV |
vector containing the first step estimates of the integrated variance of the assets, needed in the truncation. Is |
noisevar |
vector containing the estimates of the noise variance of the assets, needed in the truncation. Is |
K |
positive integer, slow time scale returns are computed on prices that are |
J |
positive integer, fast time scale returns are computed on prices that are |
KCov |
positive integer, for the extradiagonal covariance elements the slow time scale returns are computed on prices that are |
JCov |
positive integer, for the extradiagonal covariance elements the fast time scale returns are computed on prices that are |
KVar |
vector of positive integers, for the diagonal variance elements the slow time scale returns are computed on prices that are |
JVar |
vector of positive integers, for the diagonal variance elements the fast time scale returns are computed on prices that are |
eta |
positive real number, squared standardized high-frequency returns that exceed eta are detected as jumps. |
makePsd |
boolean, in case it is |
... |
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 \tau_1,\tau_2,\ldots,\tau_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_\eta^{*}}{K}\frac{\sum_{i=1}^{n-K+1}(X_{t_{i+K}}-X_{t_i})^2I_X^K(i;\eta)}{\frac{1}{n-K+1}\sum_{i=1}^{n-K+1}I_X^K(i;\eta)}.
The constant c_\eta adjusts for the bias due to the thresholding and I_{X}^K(i;\eta) is a jump indicator function
that is one if
\frac{(X_{t_{i+K}}-X_{t_{i}})^2}{(\int_{t_{i}}^{t_{i+K}} \sigma^2_sds +2\sigma_{\varepsilon_{\mbox{\tiny X}}}^2)} \ \ \leq \ \ \eta
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 \Gamma =\{ \tau_1,\tau_2,\ldots,\tau_{N^{\mbox{\tiny X}}_{\mbox{\tiny T}}}\} and
\Theta=\{\theta_1,\theta_2,\ldots,\theta_{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.
\phi_1=\max(\tau_1,\theta_1). The subsequent refresh time is defined as the first time when both stocks have again traded, i.e.
\phi_{j+1}=\max(\tau_{N^{\mbox{\tiny{X}}}_{\phi_j}+1},\theta_{N^{\mbox{\tiny{Y}}}_{\phi_j}+1}). The
complete refresh time sample grid is
\Phi=\{\phi_1,\phi_2,...,\phi_{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\{\tau\in\Gamma:\tau\leq \phi_i\} and
s_i=\max\{\theta\in\Theta:\theta\leq \phi_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{\sum_{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;\eta)
I_{Y}^K(i;\eta)}{\frac{1}{M_N-K+1}\sum_{i=1}^{M_N-K+1}{I_X^K(i;\eta)I_Y^K(i;\eta)}},
with I_{X}^K(i;\eta) 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)
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