% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/fevd.bvar.R
\name{fevd.bvar}
\alias{fevd.bvar}
\title{Forecast Error Variance Decomposition}
\usage{
\method{fevd}{bvar}(
  object,
  response = NULL,
  n.ahead = 5,
  type = "oir",
  normalise_gir = FALSE,
  period = NULL,
  ...
)
}
\arguments{
\item{object}{an object of class \code{"bvar"}, usually, a result of a call to \code{\link{bvar}}
or \code{\link{bvec_to_bvar}}.}

\item{response}{name of the response variable.}

\item{n.ahead}{number of steps ahead.}

\item{type}{type of the impulse responses used to calculate forecast error variable decompositions.
Possible choices are orthogonalised \code{oir} (default) and generalised \code{gir} impulse responses.}

\item{normalise_gir}{logical. Should the GIR-based FEVD be normalised?}

\item{period}{integer. Index of the period, for which the variance decomposition should be generated.
Only used for TVP or SV models. Default is \code{NULL}, so that the posterior draws of the last time period
are used.}

\item{...}{further arguments passed to or from other methods.}
}
\value{
A time-series object of class \code{"bvarfevd"}.
}
\description{
Produces the forecast error variance decomposition of a Bayesian VAR model.
}
\details{
The function produces forecast error variance decompositions (FEVD) for the VAR model
\deqn{A_0 y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + u_t,}
with \eqn{u_t \sim N(0, \Sigma)}. For non-structural models matrix \eqn{A_0} is set to the identiy matrix
and can therefore be omitted, where not relevant.

If the FEVD is based on the orthogonalised impulse resonse (OIR), the FEVD will be calculated as
\deqn{\omega^{OIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i P e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},}
where \eqn{\Phi_i} is the forecast error impulse response for the \eqn{i}th period,
\eqn{P} is the lower triangular Choleski decomposition of the variance-covariance
matrix \eqn{\Sigma}, \eqn{e_j} is a selection vector for the response variable and
\eqn{e_k} a selection vector for the impulse variable.

If \code{type = "sir"}, the structural FEVD will be
calculated as \deqn{\omega^{SIR}_{jk, h} = \frac{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} A_0^{-1\prime} \Phi_i^{\prime} e_j )},}
where \eqn{\sigma_{jj}} is the diagonal element of the \eqn{j}th variable of the variance covariance matrix.

If \code{type = "gir"}, the generalised FEVD will be
calculated as \deqn{\omega^{GIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i \Sigma \Phi_i^{\prime} e_j )},}
where \eqn{\sigma_{jj}} is the diagonal element of the \eqn{j}th variable of the variance covariance matrix.

If \code{type = "sgir"}, the structural generalised FEVD will be
calculated as \deqn{\omega^{SGIR}_{jk, h} = \frac{\sigma^{-1}_{jj} \sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma e_k )^2}{\sum_{i = 0}^{h-1} (e_j^{\prime} \Phi_i A_0^{-1} \Sigma A_0^{-1\prime} \Phi_i^{\prime} e_j )}}.

Since GIR-based FEVDs do not add up to unity, they can be normalised by setting \code{normalise_gir = TRUE}.
}
\examples{

# Load data
data("e1")
e1 <- diff(log(e1)) * 100

# Generate models
model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Obtain FEVD
vd <- fevd(object, response = "cons")

# Plot FEVD
plot(vd)

}
\references{
Lütkepohl, H. (2006). \emph{New introduction to multiple time series analysis} (2nd ed.). Berlin: Springer.

Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. \emph{Economics Letters, 58}, 17-29.
}