irf: Impulse Response Function

Description

Computes the impulse response coefficients of an object of class "bvar" for n.ahead steps.

A plot function for objects of class "bvarirf".

Usage

irf(object, impulse = NULL, response = NULL, n.ahead = 5,
  ci = 0.95, type = "feir", cumulative = FALSE)
# S3 method for bvarirf
plot(x, ...)

Arguments

object

an object of class "bvar", usually, a result of a call to bvar or bvec_to_bvar.

impulse

name of the impulse variable.

response

name of the response variable.

n.ahead

number of steps ahead.

a numeric between 0 and 1 specifying the probability mass covered by the credible intervals. Defaults to 0.95.

type

type of the impulse resoponse. Possible choices are forecast error "feir" (default), orthogonalised "oir", structural "sir", generalised "gir", and structural generalised "sgir" impulse responses.

cumulative

logical specifying whether a cumulative IRF should be calculated.

an object of class "bvarirf", usually, a result of a call to irf.

...

further graphical parameters.

Value

A time-series object of class "bvarirf".

Details

The function produces different types of impulse responses for the VAR model $$y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + A_0^{-1} u_t,$$ with $u_t \sim N(0, \Sigma)$.

Forecast error impulse responses $\Phi_i$ are obtained by recursions $$\Phi_i = \sum_{j = 1}^{i} \Phi_{i-j} A_j, i = 1, 2,...,h$$ with $\Phi_0 = I_K$.

Orthogonalised impulse responses $\Theta^o_i$ are calculated as $\Theta^o_i = \Phi_i P$, where P is the lower triangular Choleski decomposition of $\Sigma$. $A_0$ is assumed to be an identity matrix.

Structural impulse responses $\Theta^s_i$ are calculated as $\Theta^s_i = \Phi_i A_0^{-1}$.

(Structural) Generalised impulse responses for variable $j$, i.e. $\Theta^g_ji$ are calculated as $\Theta^g_{ji} = \sigma_{jj}^{-1/2} \Phi_i A_0^{-1} \Sigma e_j$, where $\sigma_{jj}$ is the variance of the $j^{th}$ diagonal element of $\Sigma$ and $e_i$ is a selection vector containing one in its $j^{th}$ element and zero otherwise. If the "bvar" object does not contain draws of $A_0$, it is assumed to be an identity matrix.

References

L<U+00FC>tkepohl, H. (2007). 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. Economics Letters, 58, 17-29.

Examples

Run this code

# NOT RUN {
data("e1")
e1 <- diff(log(e1))

data <- gen_var(e1, p = 2, deterministic = "const")

y <- data$Y[, 1:73]
x <- data$Z[, 1:73]

set.seed(1234567)

iter <- 500 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
store <- iter - burnin

t <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Set (uninformative) priors
a_mu_prior <- matrix(0, m) # Vector of prior parameter means
a_v_i_prior <- diag(0, m) # Inverse of the prior covariance matrix

u_sigma_df_prior <- 0 # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(.00001, k)
u_sigma <- solve(u_sigma_i)

# Data containers for posterior draws
draws_a <- matrix(NA, m, store)
draws_sigma <- matrix(NA, k^2, store)

# Start Gibbs sampler
for (draw in 1:iter) {
  # Draw conditional mean parameters
  a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior)

# Draw variance-covariance matrix
u <- y - matrix(a, k) %*% x # Obtain residuals
u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u))
u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
u_sigma <- solve(u_sigma_i) # Invert Sigma_i to obtain Sigma

# Store draws
if (draw > burnin) {
  draws_a[, draw - burnin] <- a
  draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Generate bvar object
bvar_est <- bvar(y = y, x = x, A = draws_a[1:18,],
                 C = draws_a[19:21, ], Sigma = draws_sigma)

# Generate impulse response
IR <- irf(bvar_est, impulse = "income", response = "cons", n.ahead = 8)

# Plot
plot(IR, main = "Forecast Error Impulse Response", xlab = "Period", ylab = "Response")

data("e1")
e1 <- diff(log(e1))

data <- gen_var(e1, p = 2, deterministic = "const")

y <- data$Y[, 1:73]
x <- data$Z[, 1:73]

set.seed(1234567)

iter <- 500 # Number of iterations of the Gibbs sampler
# Chosen number of iterations should be much higher, e.g. 30000.

burnin <- 100 # Number of burn-in draws
store <- iter - burnin

t <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Set (uninformative) priors
a_mu_prior <- matrix(0, m) # Vector of prior parameter means
a_v_i_prior <- diag(0, m) # Inverse of the prior covariance matrix

u_sigma_df_prior <- 0 # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(.00001, k)
u_sigma <- solve(u_sigma_i)

# Data containers for posterior draws
draws_a <- matrix(NA, m, store)
draws_sigma <- matrix(NA, k^2, store)

# Start Gibbs sampler
for (draw in 1:iter) {
  # Draw conditional mean parameters
  a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior)

# Draw variance-covariance matrix
u <- y - matrix(a, k) %*% x # Obtain residuals
u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u))
u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
u_sigma <- solve(u_sigma_i) # Invert Sigma_i to obtain Sigma

# Store draws
if (draw > burnin) {
  draws_a[, draw - burnin] <- a
  draws_sigma[, draw - burnin] <- u_sigma
  }
}

# Generate bvar object
bvar_est <- bvar(y = y, x = x, A = draws_a[1:18,],
                 C = draws_a[19:21, ], Sigma = draws_sigma)

# Generate impulse response
IR <- irf(bvar_est, impulse = "income", response = "cons", n.ahead = 8)

# Plot
plot(IR, main = "Forecast Error Impulse Response", xlab = "Period", ylab = "Response")

# }

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