llag.test: Wild Bootstrap Test for the Absence of Lead-Lag Effects

Description

Tests the absence of lead-lag effects (time-lagged correlations) by the wild bootstrap procedure proposed in Koike (2017) for each pair of components.

Usage

llag.test(x, from = -Inf, to = Inf, division = FALSE, grid, R = 999,
          parallel = "no", ncpus = getOption("boot.ncpus", 1L),
          cl = NULL, tol = 1e-06)

Value

p.values: a matrix whose components indicate the bootstrap p-values for the corresponding pair of components.
max.cov: a matrix whose componenets indicate the maxima of the absolute values of cross-covariances for the corresponding pair of components.
max.corr: a matrix whose componenets indicate the maxima of the absolute values of cross-correlations for the corresponding pair of components.

Arguments

x: an object of yuima-class or yuima.data-class.
from: a numeric vector each of whose component(s) indicates the lower end of a finite grid on which the contrast function is evaluated, if grid is missing.
to: a numeric vector each of whose component(s) indicates the upper end of a finite grid on which the contrast function is evaluated, if grid is missing.
division: a numeric vector each of whose component(s) indicates the number of the points of a finite grid on which the contrast function is evaluated, if grid is missing.
grid: a numeric vector or a list of numeric vectors. See 'Details' of llag.
R: a single positive integer indicating the number of bootstrap replicates.
parallel: passed to boot.
ncpus: passed to boot.
cl: passed to boot.
tol: tolelance parameter to avoid numerical errors in comparison of time stamps. All time stamps are divided by tol and rounded to integers. Note that the values of grid are also 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

For each pair of components, this function performs the wild bootstrap procedure proposed in Koike (2017) to test whether there is a (possibly) time-lagged correlation. The null hypothesis of the test is that there is no time-lagged correlation and the alternative is its negative. The test regects the null hypothesis if the maximum of the absolute values of cross-covariances is too large. The critical region is constructed by a wild bootstrap procedure with Rademacher variables as the multiplier variables.

References

Koike, Y. (2019). Gaussian approximation of maxima of Wiener functionals and its application to high-frequency data, Annals of Statistics, 47, 1663--1687. tools:::Rd_expr_doi("10.1214/18-AOS1731").

Examples

Run this code

if (FALSE) {
# The following example is taken from mllag

## 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)

## perform lead-lag test
llag.test(psample, grid = G, R = 999)

## Since the lead-lag parameter for the pair(x1, x3) is not contained in G,
## the null hypothesis is not rejected for this pair
}

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