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 time-series object of endogenous variables, usually, a result of a call to gen_var.

a time-series object of $(p K + (1 + s) M + N)$ regressor variables, usually, a result of a call to gen_var.

either a $K^{2} \times S$ matrix of MCMC coefficient draws of structural parameters or a named list, where element coeffs contains a $K^{2} \times S$ matrix of MCMC coefficient draws of structural parameters and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.

either a $p K^{2} \times S$ matrix of MCMC coefficient draws of lagged endogenous variables or a named list, where element coeffs contains a $p K^{2} \times S$ matrix of MCMC coefficient draws of lagged endogenous variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.

either a $((1 + s) M K) \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables or a named list, where element coeffs contains a $((1 + s) M K) \times S$ matrix of MCMC coefficient draws of unmodelled, non-deterministic variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.

either a $K N \times S$ matrix of MCMC coefficient draws of deterministic terms or a named list, where element coeffs contains a $K N \times S$ matrix of MCMC coefficient draws of deterministic terms and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.

Sigma

a $K^{2} \times S$ matrix of MCMC draws for the error variance-covariance matrix or a named list, where element coeffs contains a $K^{2} \times S$ matrix of MCMC draws for the error variance-covariance matrix and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed to the covariances.

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

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

A0_lambda

an $S \times K^{2}$ "mcmc" object of inclusion parameters for structural parameters.

an $S \times p K^{2}$ "mcmc" object of coefficient draws of lagged endogenous variables.

A_lambda

an $S \times p K^{2}$ "mcmc" object of inclusion parameters for lagged endogenous variables.

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

B_lambda

an $S \times ((1 + s) M K)$ "mcmc" object of inclusion parameters for unmodelled, non-deterministic variables.

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

C_lambda

an $S \times N K$ "mcmc" object of inclusion parameters for deterministic terms.

Sigma

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

Sigma_lambda

an $S \times K^{2}$ "mcmc" object of inclusion parameters for error covariances.

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 $u_{t} \sim N (0, Σ_{u})$ .

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

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

References

L<U+00FC>tkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Examples

Run this code

# NOT RUN {
# Get data
data("e1")
e1 <- diff(log(e1))
e1 <- window(e1, end = c(1978, 4))

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

# Add priors
model <- add_priors(data,
                    coef = list(v_i = 0, v_i_det = 0),
                    sigma = list(df = 0, scale = .00001))

# Set RNG seed for reproducibility 
set.seed(1234567)

iterations <- 400 # Number of iterations of the Gibbs sampler
# Chosen number of iterations and burnin should be much higher.
burnin <- 100 # Number of burn-in draws
draws <- iterations + burnin # Total number of MCMC draws

y <- t(model$data$Y)
x <- t(model$data$Z)
tt <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
m <- k * nrow(x) # Number of estimated coefficients

# Priors
a_mu_prior <- model$priors$coefficients$mu # Vector of prior parameter means
a_v_i_prior <- model$priors$coefficients$v_i # Inverse of the prior covariance matrix

u_sigma_df_prior <- model$priors$sigma$df # Prior degrees of freedom
u_sigma_scale_prior <- model$priors$sigma$scale # Prior covariance matrix
u_sigma_df_post <- tt + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
u_sigma_i <- diag(1 / .00001, k)

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

# Start Gibbs sampler
for (draw in 1:draws) {
 # 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)

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

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

# Load data
data("e1")
e1 <- diff(log(e1)) * 100
e1 <- window(e1, end = c(1978, 4))

# Generate model data
model <- gen_var(e1, p = 2, deterministic = 2,
                 iterations = 100, burnin = 10)
# Chosen number of iterations and burnin should be much higher.

# Add prior specifications
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Generate forecasts
bvar_pred <- predict(object, n.ahead = 10, new_d = rep(1, 10))

# Plot forecasts
plot(bvar_pred)

# }

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