R/g.fevd.R

In Spillover: Spillover/Connectedness Index Based on VAR Modelling

#' Generalized Forecast Error Variance Decomposition
#' 
#'@description
#' Computes the generalized forecast error variance decomposition of a VAR(p) for \code{n.ahead} steps.
#' 
#' @param x Object of class \sQuote{\code{varest}} generated by \code{VAR()} from vars package.
#' @param n.ahead Integer specifying the steps ahead.
#' @param normalized a logical value indicating whether the result should be normalized to sum up to 1, see Details 
#' 
#' @details 
#' When \code{normalized=FALSE} this function computes the generalized forecast error variance decomposition proposed by Pesaran and Shin (1998) which takes the form:
#'  \deqn{
#'      \alpha _{ij}^{g}(h) = \frac{\sigma_{ii}^{-1}\sum_{l=0}^{h-1}(\mathbf{e'}_{i}\Theta _{l}\Sigma_{\varepsilon}\mathbf{e}_{j})^{2}}{\sum_{l=0}^{h-1}(\mathbf{e'}_{i}\Theta _{l}\Sigma _{\varepsilon }\Theta'_{l}\mathbf{e}_{i})}, \quad i,j = 0,1,2\ldots, K
#'      }{
#'      \alpha_ij^g(h) = \sigma_ii^-1 \sum_l=0^h-1 (e'_i \Theta_l \Sigma_\epsilon e_j)^2 / \sum_l=0^h-1 (e'_i \Theta_l \Sigma_\epsilon \Theta'_l e_i),       i,j = 0,1,2 ..., K 
#'      }
#'      
#'  Where \eqn{\mathbf{\Theta}_{l}}{\Theta_l}, are the coefficients matrix of the MA representation of the VAR model, \eqn{\mathbf{\Sigma}_{\varepsilon}}{\Sigma_\epsilon} 
#'  is the variance matrix of the reduced-form error vector \eqn{\varepsilon}{\epsilon}, \eqn{\sigma_{ii}}{\sigma_ii} is the standard deviation of the error term for the 
#'  \eqn{ith}{ith} equation and \eqn{e_{i}}{e_i} and \eqn{e_{j}}{e_j} are selection vectors with ones as the \emph{ith} element and zeros elsewhere. 
#'    
#'  If \code{normalized=TRUE} (the default value) then \code{g.fevd} computes:
#'  \deqn{
#'  \tilde{a}_{ij}^{g}(h) = \frac{a_{ij}^{g}(h)} {\sum_{j=1}^{K} a_{ij}^{g}(h)} 
#'  }{
#'  \tilde{a}_ij^g(h) = a_ij^g(h) / \sum_j=1^K a_ij^g(h) 
#'  } 
#'  This fact implies the normalization is simply each entry of the generalized fevd divided by the its corresponding row sum.
#'
#'@return A list of length \emph{K} holding the generalized forecast error variances as matrices. This is an object of class \sQuote{\code{varfevd}} from vars package.
#'
#'@references 
#'Pesaran, M. H. and Shin, Y. (1998). \emph{Generalized impulse response analysis in linear multivariate models}. Economics Letters, 58(1):17-29.
#'@author Jilber Urbina
#'@export
#'@examples 
#' library(vars)
#' data(stock.prices)
#' stocks <- stock.prices[,1:2]
#' VAR.1 <- VAR(stocks)
#' g.fevd(VAR.1, n.ahead = 10) # normalized 
#' g.fevd(VAR.1, n.ahead = 10, normalized=FALSE) # Not normalized 
#' @keywords
#' fevd
#' g.fevd
#' Generalized error variance decomposition



g.fevd <- function (x, n.ahead = 10, normalized = TRUE) 
{
  if (!inherits(x, "varest")) {
    stop("\nPlease provide an object of class 'varest', generated by 'VAR()'.\n")
  }
  
  Sigma <-  summary(x)$covres
  n.ahead <- abs(as.integer(n.ahead))
  A <- vars::Phi(x, n.ahead-1)
  sigma_ii <- diag(1/sqrt(diag(Sigma)))
  Sigma_A <- vector("list", n.ahead)
  A_Sigma_A <- vector("list", n.ahead)
  
  for (h in 1:n.ahead) {
    # Para el numerador
    Sigma_A[[h]] <-  (A[,,h] %*% Sigma %*% sigma_ii)^2
    
    # Para denominador
    A_Sigma_A[[h]] <- A[,,h] %*% Sigma %*% t(A[,,h])
  }
  
  
  # Numerador
  num <- Reduce("+", Sigma_A, accumulate = TRUE)
  
  # Denominador
  den <- Reduce("+", A_Sigma_A, accumulate = TRUE)
  
  gfevd.unordered <- lapply(1:n.ahead, function(h) num[[h]]/diag(den[[h]]))
  
  # Una función para ordenar
  bind.ith.rows <- function(i) do.call(rbind, lapply(gfevd.unordered, "[", i, TRUE))
  
  nr <- nrow(gfevd.unordered[[1]])
  gfevd <- lapply(1:nr, bind.ith.rows)
  
  if (normalized) {
    gfevd <- lapply(gfevd, function(x) {
      sweep(x, 1, STATS = rowSums(x), FUN = "/")
    })
    names(gfevd) <- colnames(Sigma)
    
  }
  else {
    names(gfevd) <- colnames(Sigma)
    
  }
  
  gfevd <- lapply(gfevd, function(elements){
    colnames(elements) <- colnames(Sigma)
    elements
  })
  
  class(gfevd) <- "varfevd"
  return(gfevd)
  
}  ## End