yuima (version 1.15.27)

wllag: Scale-by-scale lead-lag estimation

Description

This function estimates lead-lag parameters on a scale-by-scale basis from non-synchronously observed bivariate processes, using the estimatiors proposed in Hayashi and Koike (2018b).

Usage

wllag(x, y, J = 8, N = 10, tau = 1e-3, from = -to, to = 100, 
      verbose = FALSE, in.tau = FALSE, tol = 1e-6)

Value

If verbose is FALSE, a numeric vector with length J, corresponding to the estimated scale-by-scale lead-lag parameters, is returned. Note that their positive values indicate that the first process leads the second process.

Otherwise, an object of class "yuima.wllag", which is a list with the following components, is returned:

lagtheta

the estimated scale-by-scale lead-lag parameters. The \(j\) th component corresponds to the estimate at the level \(j\). A positive value indicates that the first process leads the second process.

obj.values

the values of the objective functions evaluated at the estimated lead-lag parameters.

obj.fun

a list of values of the objective functions. The \(j\) th component of the list corresponds to a zoo object for values of the signed objective function at the level \(j\) indexed by the search grid.

theta.hry

the lead-lag parameter estimate in the sense of Hoffmann, Rosenbaum and Yoshida (2013).

cor.hry

the correltion coefficient in the sense of Hoffmann, Rosenbaum and Yoshida (2013), evaluated at the estimated lead-lag parameter.

ccor.hry

a zoo object for values of the cross correltion function in the sense of Hoffmann, Rosenbaum and Yoshida (2013) indexed by the search grid.

Arguments

x

a zoo object for observation data of the first process.

y

a zoo object for observation data of the second process.

J

a positive integer. Scale-by scale lead-lag parameters are estimated up to the level J.

N

The number of vanishing moments of Daubechies' compactly supported wavelets. This should be an integer between 1 and 10.

tau

the step size of a finite grid on which objective functions are evaluated. Note that this value is identified with the finest time resolution of the underlying model. The default value 1e-3 corresponds to 1 mili-second if the unit time corresponds to 1 second.

from

a negative integer. from*tau gives the lower end of a finite grid on which objective functions are evaluated.

to

a positive integer. to*tau gives the upper end of a finite grid on which objective functions are evaluated.

verbose

a logical. If FALSE (default), the function returns only the estimated scale-by-scale lead-lag parameters. Otherwise, the function also returns some other statistics such as values of the signed objective functions. See `Value'.

in.tau

a logical. If TRUE, the estimated lead-lag parameters are returned in increments of tau. That is, the estimated lead-lag parameters are divided by tau.

tol

tolelance parameter to avoid numerical errors in comparison of time stamps. All time stamps are divided by tol and rounded to integers. A reasonable choice of tol is the minimum unit of time stamps. The default value 1e-6 supposes that the minimum unit of time stamps is greater or equal to 1 micro-second.

Author

Yuta Koike with YUIMA Project Team

Details

Hayashi and Koike (2018a) introduced a bivariate continuous-time model having different lead-lag relationships at different time scales. The wavelet cross-covariance functions of this model, computed based on the Littlewood-Paley wavelets, have unique maximizers in absolute values at each time scale. These maximizer can be considered as lead-lag parameters at each time scale. To estimate these parameters from discrete observation data, Hayashi and Koike (2018b) constructed objective functions mimicking behavior of the wavelet cross-covariance functions of the underlying model. Then, estimates of the scale-by-scale lead-lag parameters can be obtained by maximizing these objective functions in absolute values.

References

Hayashi, T. and Koike, Y. (2018a). Wavelet-based methods for high-frequency lead-lag analysis, SIAM Journal of Financial Mathematics, 9, 1208--1248.

Hayashi, T. and Koike, Y. (2018b). Multi-scale analysis of lead-lag relationships in high-frequency financial markets. tools:::Rd_expr_doi("10.48550/arXiv.1708.03992").

Hoffmann, M., Rosenbaum, M. and Yoshida, N. (2013) Estimation of the lead-lag parameter from non-synchronous data, Bernoulli, 19, no. 2, 426--461.

See Also

simBmllag, llag

Examples

if (FALSE) {
## An example from a simulation setting of Hayashi and Koike (2018b)
set.seed(123)

# Simulation of Bm driving the log-price processes
n <- 15000
J <- 13
tau <- 1/2^(J+1)

rho <- c(0.3,0.5,0.7,0.5,0.5,0.5,0.5,0.5)
theta <- c(-1,-1, -2, -2, -3, -5, -7, -10) * tau

dB <- simBmllag(n, J, rho, theta)

Time <- seq(0, by = tau, length.out = n) # Time index
x <- zoo(diffinv(dB[ ,1]), Time) # simulated path of the first process
y <- zoo(diffinv(dB[ ,2]), Time) # simulated path of the second process

# Generate non-synchronously observed data
x <- x[as.logical(rbinom(n + 1, size = 1, prob = 0.5))]
y <- y[as.logical(rbinom(n + 1, size = 1, prob = 0.5))]

# Estimation of scale-by-scale lead-lag parameters (compare with theta/tau)
wllag(x, y, J = 8, tau = tau, tol = tau, in.tau = TRUE)
# Estimation with other information
out <- wllag(x, y, tau = tau, tol = tau, in.tau = TRUE, verbose = TRUE)
out

# Plot of the HRY cross-correlation function
plot(out$ccor.hry, xlab = expression(theta), ylab = expression(U(theta))) 
dev.off()

# Plot of the objective functions
op <- par(mfrow = c(4,2))
plot(out)
par(op)
}