mllag: Multiple Lead-Lag Detector

Description

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).

Usage

mllag(x, from = -Inf, to = Inf, division = FALSE, grid, psd = TRUE, 
      plot = TRUE, alpha = 0.01, fisher = TRUE, bw)

Value

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.

Arguments

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 posive 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.

Author

Yuta Koike with YUIMA Project Team

Details

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 possiblity that no lead-lag is returned). Note that this approach is mathematically debetable 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.

References

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.

Examples

Run this code


# 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!

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