bvar: Bayesian Vector Autoregression Objects

Description

bvar is used to create objects of class "bvar".

Forecasting a Bayesian VAR object of class "bvar" with credible bands.

Usage

bvar(data = NULL, exogen = NULL, y = NULL, x = NULL, A0 = NULL,
  A = NULL, B = NULL, C = NULL, Sigma = NULL)
# S3 method for bvar
predict(object, ..., n.ahead = 10, new_x = NULL,
  new_D = NULL, ci = 0.95)

Arguments

data

the original time-series object of endogenous variables.

exogen

the original time-series object of unmodelled variables.

a $K \times T$ matrix of endogenous variables, usually, a result of a call to gen_var.

a $(pK + (1+s)M + N) \times T$ matrix of regressor variables, usually, a result of a call to gen_var.

a $K^2 \times S$ matrix of MCMC coefficient draws of structural parameters.

a $pK^2 \times S$ matrix of MCMC coefficient draws of lagged endogenous variables.

a $((1 + s)MK) \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables.

an $KN \times S$ matrix of MCMC coefficient draws of deterministic terms.

Sigma

a $K^2 \times S$ matrix of variance-covariance MCMC draws.

object

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

...

additional arguments.

n.ahead

number of steps ahead at which to predict.

new_x

a matrix of new non-deterministic, exogenous variables. Must have n.ahead rows.

new_D

a matrix of new deterministic variables. Must have n.ahead rows.

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

Value

An object of class "bvar" containing the following components, if specified:

data

the original time-series object of endogenous variables.

exogen

the original time-series object of unmodelled variables.

a $K \times T$ matrix of endogenous variables.

a $(pK + (1+s)M + N) \times T$ matrix of regressor variables.

an $S \times K^2$ "mcmc" object of coefficient draws of structural parameters.

an $S \times pK^2$ "mcmc" object of coefficient draws of lagged endogenous variables.

an $S \times ((1 + s)MK)$ "mcmc" object of coefficient draws of unmodelled, non-deterministic variables.

an $S \times NK$ "mcmc" object of coefficient draws of deterministic terms.

Sigma

an $S \times K^2$ "mcmc" object of variance-covariance draws.

specifications

a list containing information on the model specification.

A time-series object of class "bvarprd".

Details

For the VARX model $$A_0 y_t = \sum_{i = 1}^{p} A_i y_{t-i} + \sum_{i = 0}^{s} B_i x_{t - i} + C d_t + u_t$$ the function collects the S draws of a Gibbs sampler (after the burn-in phase) in a standardised object, where $y_t$ is a K-dimensional vector of endogenous variables, $A_0$ is a $K \times K$ matrix of structural coefficients. $A_i$ is a $K \times K$ coefficient matrix of lagged endogenous variabels. $x_t$ is an M-dimensional vector of unmodelled, non-deterministic variables and $B_i$ its corresponding coefficient matrix. $d_t$ is an N-dimensional vector of deterministic terms and $C$ its corresponding coefficient matrix. $u_t$ is an error term with $u_t \sim N(0, \Sigma_u)$.

The draws of the different coefficient matrices provided in A0, A, B, C and Sigma have to correspond to the same MCMC iteration.

For the VAR model $$y_t = \sum_{i = 1}^{p} A_{i} y_{t-i} + \sum_{i = 0}^{s} B_{i} x_{t-i} + C D_t + A_0^{-1} u_t,$$ with $u_t \sim N(0, \Sigma)$ the function produces n.ahead forecasts.

References

L<U+00FC>tkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

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)
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 forecasts
bvar_pred <- predict(bvar_est, n.ahead = 10, new_D = rep(1, 10))

# Plot forecasts
plot(bvar_pred)

# }

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