irf {bvartools}R Documentation

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 class '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.

ci

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.

x

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

...

further graphical parameters.

Details

The function produces different types of impulse responses for the VAR model

y_t = ∑_{i = 1}^{p} A_{i} y_{t-i} + A_0^{-1} u_t,

with u_t \sim N(0, Σ).

Forecast error impulse responses Φ_i are obtained by recursions

Φ_i = ∑_{j = 1}^{i} Φ_{i-j} A_j, i = 1, 2,...,h

with Φ_0 = I_K.

Orthogonalised impulse responses Θ^o_i are calculated as Θ^o_i = Φ_i P, where P is the lower triangular Choleski decomposition of Σ. A_0 is assumed to be an identity matrix.

Structural impulse responses Θ^s_i are calculated as Θ^s_i = Φ_i A_0^{-1}.

(Structural) Generalised impulse responses for variable j, i.e. Θ^g_ji are calculated as Θ^g_{ji} = σ_{jj}^{-1/2} Φ_i A_0^{-1} Σ e_j, where σ_{jj} is the variance of the j^{th} diagonal element of Σ 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.

Value

A time-series object of class "bvarirf".

References

Lütkepohl, 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

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")


[Package bvartools version 0.0.2 Index]