R : VAR Impulse Response Function

This post draws the orthogonal impulse response functions from an estimated VAR model with 1 standard deviation or 1 unit shock. vars R package does not provide 1 unit shock so I implemented it using the Cholesky decomposition.

Orthogonal IRF of VAR model

R code

The following R code loads yield curve data, calculates empirical yield factors, and estimates VAR(1) model using vars R package.

#========================================================# # Quantitative Financial Econometrics & Derivatives  # Machine/Deep Learing, R, Python, Tensorflow  # by Sang-Heon Lee  # # https://shleeai.blogspot.com #--------------------------------------------------------# # Orthogonal Impulse Response #========================================================# graphics.off(); rm(list = ls())   library(vars)   #----------------------------------------------------------- # Data : US Yield Curve 1972 - 2000 #----------------------------------------------------------- url <- 'https://www.dropbox.com/s/inpnlugzkddp42q/bonds.csv?dl=1' df  <- read.csv(url, sep = ";")   # Empirical yield factors L <- df[,"M120"] S <- df[,"M3"] - df[,"M120"] C <- 2 * df[,"M30"] - df[,"M3"] - df[,"M120"]   # VAR endogenous variables data  <- data.frame(L=L, S=S, C=C) nvar  <- ncol(data) nobs  <- nrow(data) vname <- colnames(data) head(data)   #----------------------------------------------------------- # VAR estimation #----------------------------------------------------------- var.est <- VAR(data, p = 1, type = "const", season=NULL) var.summary <- summary(var.est) var.summary   Colored by Color Scripter

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> head(data)       L      S     C 1 6.088 -2.706 0.730 2 6.283 -2.813 0.327 3 6.269 -2.395 1.037 4 6.240 -2.592 0.820 5 6.249 -2.414 0.272 6 6.301 -2.200 0.710

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The orthogonal impulse responses with 1 SD (standard deviation) shocks are easily calculated using the irf() function as follows.

#----------------------------------------------------------- # one SD shock IRF #----------------------------------------------------------- irf_1sd <- list() for (im in 1:nvar) {     irf_1sd[[im]] <- irf(var.est, impulse = vname[im],                           n.ahead = 36,                           ortho = TRUE, boot = TRUE)     windows(width = 4, height = 6)     plot(irf_1sd[[im]], lwd = c(3,1,1), col = "black",           main = paste0("1 SD orthogonal IRF from ", vname[im])) }   Colored by Color Scripter

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However, as far as I can tell, the irf() function does not provide the 1 unit shock IRFs. The IRFs based on 1 SD shock or 1 unit shock have the same shape, just different scales. In the latter case, the size of the shock is 1 unit, and since the unit of the variable used in this post is %, the 1 unit shock corresponds to the 1% shock. I calculate it by using the Cholesky decomposition as follows.

#----------------------------------------------------------- # one unit shock IRF #-----------------------------------------------------------   # Cholesky decomposition for scaling # PP' = cov P = t(chol(var.summary$covres)) vscale <- diag(P); vscale   irf_1ut <- list()  for (im in 1:nvar) {       irf_1ut[[im]] <- irf_1sd[[im]]          # To make shock size 1 unit, dividing by diag(P)      irf_1ut[[im]]$irf[[1]]   <- irf_1ut[[im]]$irf[[1]]/vscale[im]     irf_1ut[[im]]$Lower[[1]] <- irf_1ut[[im]]$Lower[[1]]/vscale[im]     irf_1ut[[im]]$Upper[[1]] <- irf_1ut[[im]]$Upper[[1]]/vscale[im]          windows(width = 4, height = 6)     plot(irf_1ut[[im]], lwd = c(3,1,1), col = "blue",          main = paste0("1 unit orthogonal IRF from ", vname[im])) }   Colored by Color Scripter

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It would be nice if the IRF plots generated by the irf() function were displayed in a matrix like EViews for multiple impulse shocks, but this is not the case. Therefore, we need to concatenate mannualy the IRF plots, which appear as a single column, horizontally,

In the next post, I'd like to discuss the IRF using the local projection of Jordà (2005), which takes nonlinearities into account to calculate IRFs.