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#' Sample cross extremogram
#'
#' @description The function estimates the sample cross extremogram and creates an extremogram plot.
#' @param a Bivariate time series (n by 2 matrix).
#' @param quant1 Quantile of the first time series to indicate an extreme event (a number between 0 and 1).
#' @param quant2 Quantile of the second time series to indicate an extreme event (a number between 0 and 1).
#' @param maxlag Number of lags to include in the extremogram (an integer).
#' @param type If type=1, the upper tail extremogram is estimated - P(Y>y,X>x).
#' If type=2, the lower tail extremogram is estimated - P(Y<y,X<x).
#' If type=3, the extremogram is estimated for a lower tail extreme value in the
#' first time series and an upper tail extreme value in the second time series - P(Y>y,X<x).
#' If type=4, the extremogram is estimated for a lower tail extreme value in the
#' second time series and an upper tail extreme value in the first time series - P(Y<y,X>x).
#' @param ploting An extremogram plot. If ploting = 1, a plot is created (default). If ploting = 0,
#' no plot is created.
#' @param cutoff The cutoff of the y-axis on the plot (a number between 0 and 1, default is 1).
#' @param start The lag that the extremogram plots starts at (an integer not greater than \code{maxlag}, default is 0).
#' @return Cross extremogram values and a plot (if requested).
#' @references \enumerate{
#' \item Davis, R. A., Mikosch, T., & Cribben, I. (2012). Towards estimating extremal
#' serial dependence via the bootstrapped extremogram. Journal of Econometrics,170(1),
#' 142-152.
#' \item Davis, R. A., Mikosch, T., & Cribben, I. (2011). Estimating extremal
#' dependence in univariate and multivariate time series via the extremogram.arXiv
#' preprint arXiv:1107.5592.}
#' @examples
#' # generate a GARCH(1,1) process
#' omega = 1
#' alpha1 = 0.1
#' beta1 = 0.6
#' alpha2 = 0.11
#' beta2 = 0.78
#' n = 1000
#' quant = 0.95
#' type = 1
#' maxlag = 70
#' df = 3
#' G1 = extremogram:::garchsim(omega,alpha1,beta1,n,df)
#' G2 = extremogram:::garchsim(omega,alpha2,beta2,n,df)
#' data = cbind(G1, G2)
#'
#' extremogram2(data, quant, quant, maxlag, type, 1, 1, 0)
#' @export
extremogram2 = function(a, quant1, quant2, maxlag, type, ploting=1, cutoff=1, start=0) {
x=a[,1]
y=a[,2]
level1 = quantile(a[,1],prob = quant1);level2 = quantile(a[,2],prob = quant2)
n = length(a[,1]); rhohat = rep(0,maxlag)
if (type == 1) {
for ( i in 1:maxlag) {
rhohat[i]=length((1:(n-i))[x[1:(n-i+1)] > level1 & y[i:n]> level2])
rhohat[i]=rhohat[i]/length((1:(n-i))[x[1:(n-i+1)]>level1])
}
}
else
if (type == 2) {
for ( i in 1:maxlag) {
rhohat[i]=length((1:(n-i))[x[1:(n-i+1)] < level1 & y[i:n]< level2])
rhohat[i]=rhohat[i]/length((1:(n-i))[x[1:(n-i+1)] < level1])
}
}
else
if (type == 3) {
for ( i in 1:maxlag) {
rhohat[i]=length((1:(n-i))[x[1:(n-i+1)] < level1 & y[i:n]> level2])
rhohat[i]=rhohat[i]/length((1:(n-i))[x[1:(n-i+1)] < level1])
}
}
else
if (type == 4) {
for ( i in 1:maxlag) {
rhohat[i]=length((1:(n-i))[x[1:(n-i+1)] > level1 & y[i:n]< level2])
rhohat[i]=rhohat[i]/length((1:(n-i))[x[1:(n-i+1)] > level1])
}
}
if (ploting == 1) {
plot((start:(maxlag-1)),rhohat[(start+1):maxlag],type="n",xlab="lag",ylab="extremogram",ylim=c(0,cutoff))
lines((start:(maxlag-1)),rhohat[(start+1):maxlag],col=1,lwd=1,type="h")
abline((0:(maxlag-1)),0,col=1,lwd=1)
}
return(rhohat)
}
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