"Kronid" <- function(x,plag=10,crit=0.05){
# Identifies the Kronecker indexes for a vector time series
# plag is the number of lags used to represent the PAST vector
# developed by R. Tsay in May, 2007
T=dim(x)[1]
k=dim(x)[2]
y=as.matrix(x)
if(plag < 1){
plag=floor(log(T))+1
}
# Construct the PAST-vector
iend=T-plag
past=y[1:iend,]
if (plag > 1){
for (i in 2:plag){
past=cbind(y[i:(iend+i-1),],past)
}
}
# initialize the Kronecker indexes and control variable.
kdx=rep(0,k)
found=rep(0,k)
# futu: denotes the FUTURE-vector, its dimension is increasing.
h=0
ist=plag+1
futu1=as.matrix(y[ist:T,])
print("h")
print(h)
for (i in 1:k){
print("component")
print(i)
s1=c(i)
if(i > 1){
fnd=found[1:(i-1)]
jdx=c(1:(i-1))[fnd==0]
s1=c(jdx,i)
}
futu=as.matrix(futu1[,s1])
m1=cancor(past,futu)
df=dim(futu)[2]
dp=dim(past)[2]
deg=dp-df+1
print("square of the smallest can. corr.")
seig=m1$cor[df]^2
print(seig)
tst=-(T-1-0.5*(dp+df-1))*log(1-seig)
pv=1-pchisq(tst,deg)
stat=c(tst,deg,pv)
print("test,  df, & p-value")
print(stat,digits=4)
if(i>1){
cstar=cbind(cstar,stat)
}
else{
cstar=stat
}
if(pv > crit){
found[i]=1
kdx[i]=h
print("A Kronecker index found")
}
}
print("=============")
while(sum(found) < k){
idim=dim(past)[1]
h=h+1
print("h")
print(h)
past=past[1:(idim-1),]
futu=futu[1:(idim-1),]
futu1=y[(ist+h):T,]
for (ii in 1:k){
if(found[ii]==0){
print("component")
print(ii)
futu=cbind(futu,futu1[,ii])
m1=cancor(past,futu)
df=dim(futu)[2]
dp=dim(past)[2]
deg=dp-df+1
print("Square of the smallest can. corr.")
seig=m1$cor[df]^2
print(seig)
y1=futu%*%(m1$ycoef[,df])
x1=past%*%(m1$xcoef[,df])
m2=acf(y1,lag=h,plot=F)
acfy=m2$acf[2:(h+1)]
m3=acf(x1,lag=h,plot=F)
acfx=m3$acf[2:(h+1)]
dsq=1+2*sum(acfx*acfy)
seig=seig/dsq
tst=-(T-1-0.5*(dp+df-1))*log(1-seig)
pv=1-pchisq(tst,deg)
stat=c(tst,deg,pv,dsq)
print("test,   df, p-value & dsq")
print(stat,digits=4)
stat=stat[1:3]
cstar=cbind(cstar,stat)
if(pv > crit){
found[ii]=1
kdx[ii]=h
futu=futu[,1:(df-1)]
print("A Kronecker found")
}
}
}
print("============")
}
print("            ")
print("Kronecker indexes identified:")
print(kdx)

Kronid<-list(index=kdx,tests=cstar)
}