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##' A function used to obtain partial cross-correlation function for a give lag order
##'
##' This function obtains the partial corss-correlation and the simple correlation.
##' To obtain the partial cross-correlation, this function uses the first column of
##' the input matrix and k-lagged values of the rest of the matrix.
##' @title Partial Cross-correlation function
##' @param matH A matrix with multiple columns (more than 3 columns)
##' @param k The lag order (integer)
##' @return Partial corss-correlation at k lags and the correlation statistics at k lags.
##'
##' @author Heejoon Han, Oliver Linton, Tatsushi Oka and Yoon-Jae Whang
corr.lag.partial = function(matH, k)
{
## size
Tsize = nrow(matH) ## =: T
Nvar = ncol(matH) ## =: #var
Nsize = Tsize - k
## {H_1t, H_2t-k}
matD = matrix(0, Nsize, Nvar)
matD[,1] = as.matrix(matH[(k+1):Tsize, 1, drop=FALSE])
matD[,2:Nvar] = as.matrix(matH[ 1:Nsize, 2:Nvar, drop=FALSE])
## the following matrix contains inner-products of two vectors in H.
matDD = t(matD) %*% matD
## cross-quantilogram of lag order k
CRQ = matDD[1,2] / sqrt( matDD[1,1] * matDD[2,2] ) ## 1 x 1
## partial quantilogram
invDD = solve(matDD)
ParCRQ = - invDD[1,2] / sqrt( invDD[1,1] * invDD[2,2] )
## list
list(CRQ = CRQ, ParCRQ = ParCRQ)
} ## EoF
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