Multiple Lead-Lag Detector
Detecting the lead-lag parameters of discretely observed processes by picking time shifts at which the Hayashi-Yoshida cross-correlation functions exceed thresholds, which are constructed based on the asymptotic theory of Hayashi and Yoshida (2011).
mllag(x, from = -Inf, to = Inf, division = FALSE, grid, psd = TRUE, plot = TRUE, alpha = 0.01, fisher = TRUE, bw)
x: an object of yuima-class or yuima.data-class or yuima.llag-class (output of llag) or yuima.mllag-class (output of this function).from: passed to llag.to: passed to llag.division: passed to llag.grid: passed to llag.psd: passed to llag.plot: logical. If TRUE, the estimated cross-correlation functions and the pointwise confidence intervals (under the null hypothesis that the corresponding correlation is zero) as well as the detected lead-lag parameters are plotted.alpha: a positive number indicating the significance level of the confidence intervals for the cross-correlation functions.fisher: logical. If TRUE, the p-values and the confidence intervals for the cross-correlation functions is evaluated after applying the Fisher z transformation.bw: passed to llag.The computation method of cross-correlation functions and confidence intervals is the same as the one used in llag. The exception between this function and llag is how to detect the lead-lag parameters. While llag only returns the maximizer of the absolute value of the cross-correlations following the theory of Hoffmann et al. (2013), this function returns all the time shifts at which the cross-correlations exceed (so there is also the possibility that no lead-lag is returned). Note that this approach is mathematically debatable because there would be a multiple testing problem (see also 'Note' of llag), so the interpretation of the result from this function should carefully be addressed. In particular, the significance level alpha probably does not give the "correct" level.
An object of class "yuima.mllag", which is a list with the following elements: - mlagcce: a list of data.frame-class objects consisting of lagcce (lead-lag parameters), p.value and correlation.
LLR: a matrix consisting of lead-lag ratios. See Huth and Abergel (2014) for details.
ccor: a list of computed cross-correlation functions.
avar: a list of computed asymptotic variances of the cross-correlations (if ci = TRUE).
CI: a list of computed confidence intervals.
Hayashi, T. and Yoshida, N. (2011) Nonsynchronous covariation process and limit theorems, Stochastic processes and their applications, 121 , 2416--2454.
Hoffmann, M., Rosenbaum, M. and Yoshida, N. (2013) Estimation of the lead-lag parameter from non-synchronous data, Bernoulli, 19 , no. 2, 426--461.
Huth, N. and Abergel, F. (2014) High frequency lead/lag relationships --- Empirical facts, Journal of Empirical Finance, 26 , 41--58.
Yuta Koike with YUIMA Project Team
llag, hyavar, llag.test
# The first example is taken from llag ## Set a model diff.coef.matrix <- matrix(c("sqrt(x1)", "3/5*sqrt(x2)", "1/3*sqrt(x3)", "", "4/5*sqrt(x2)","2/3*sqrt(x3)", "","","2/3*sqrt(x3)"), 3, 3) drift <- c("1-x1","2*(10-x2)","3*(4-x3)") cor.mod <- setModel(drift = drift, diffusion = diff.coef.matrix, solve.variable = c("x1", "x2","x3")) set.seed(111) ## We use a function poisson.random.sampling ## to get observation by Poisson sampling. yuima.samp <- setSampling(Terminal = 1, n = 1200) yuima <- setYuima(model = cor.mod, sampling = yuima.samp) yuima <- simulate(yuima,xinit=c(1,7,5)) ## intentionally displace the second time series data2 <- yuima@data@zoo.data[[2]] time2 <- time(data2) theta2 <- 0.05 # the lag of x2 behind x1 stime2 <- time2 + theta2 time(yuima@data@zoo.data[[2]]) <- stime2 data3 <- yuima@data@zoo.data[[3]] time3 <- time(data3) theta3 <- 0.12 # the lag of x3 behind x1 stime3 <- time3 + theta3 time(yuima@data@zoo.data[[3]]) <- stime3 ## sampled data by Poisson rules psample<- poisson.random.sampling(yuima, rate = c(0.2,0.3,0.4), n = 1000) ## We search lead-lag parameters on the interval [-0.1, 0.1] with step size 0.01 G <- seq(-0.1,0.1,by=0.01) ## lead-lag estimation by mllag par(mfcol=c(3,1)) result <- mllag(psample, grid = G) ## Since the lead-lag parameter for the pair(x1, x3) is not contained in G, ## no lead-lag parameter is detected for this pair par(mfcol=c(1,1)) # The second example is a situation where multiple lead-lag effects exist set.seed(222) n <- 3600 Times <- seq(0, 1, by = 1/n) R1 <- 0.6 R2 <- -0.3 dW1 <- rnorm(n + 10)/sqrt(n) dW2 <- rnorm(n + 5)/sqrt(n) dW3 <- rnorm(n)/sqrt(n) x <- zoo(diffinv(dW1[-(1:10)] + dW2[1:n]), Times) y <- zoo(diffinv(R1 * dW1[1:n] + R2 * dW2[-(1:5)] + sqrt(1- R1^2 - R2^2) * dW3), Times) ## In this setting, both x and y have a component leading to the other, ## but x's leading component dominates y's one yuima <- setData(list(x, y)) ## Lead-lag estimation by llag G <- seq(-30/n, 30/n, by = 1/n) est <- llag(yuima, grid = G, ci = TRUE) ## The shape of the plotted cross-correlation is evidently bimodal, ## so there are likely two lead-lag parameters ## Lead-lag estimation by mllag mllag(est) # succeeds in detecting two lead-lag parameters ## Next consider a non-synchronous sampling case psample <- poisson.random.sampling(yuima, n = n, rate = c(0.8, 0.7)) ## Lead-lag estimation by mllag est <- mllag(psample, grid = G) est # detects too many lead-lag parameters ## Using a lower significant level mllag(est, alpha = 0.001) # insufficient ## As the plot reveals, one reason is because the grid is too dense ## In fact, this phenomenon can be avoided by using a coarser grid mllag(psample, grid = seq(-30/n, 30/n, by=5/n)) # succeeds!