This vignette provides the code to set up and estimate a basic Bayesian vector error correction (BVEC) model with the bvartools
package. The presented Gibbs sampler is based on the approach of Koop et al. (2010), who propose a prior on the cointegration space.
To illustrate the estimation process, the dataset E6 from Lütkepohl (2007) is used, which contains data on German long-term interest rates and inflation from 1972Q2 to 1998Q4.
library(bvartools)
data("e6")
plot(e6) # Plot the series
The gen_vec
function produces the inputs Y
, W
and X
for the BVEC estimator, where Y
is the matrix of dependent variables, W
is a matrix of potentially cointegrated regressors, and X
is the matrix of non-cointegration regressors.
data <- gen_vec(e6, p = 4, const = "unrestricted", season = "unrestricted")
y <- data$Y
w <- data$W
x <- data$X
# Reset random number generator for reproducibility
set.seed(1234567)
iter <- 10000 # Number of iterations of the Gibbs sampler
burnin <- 5000 # Number of burn-in draws
store <- iter - burnin
r <- 1 # Set rank
t <- ncol(y) # Number of observations
k <- nrow(y) # Number of endogenous variables
k_w <- nrow(w) # Number of regressors in error correction term
k_x <- nrow(x) # Number of differenced regressors and unrestrictec deterministic terms
k_alpha <- k * r # Number of elements in alpha
k_beta <- k_w * r # Number of elements in beta
k_gamma <- k * k_x
# Set uninformative priors
a_mu_prior <- matrix(0, k_x * k) # Vector of prior parameter means
a_v_i_prior <- diag(0, k_x * k) # Inverse of the prior covariance matrix
v_i <- 0
p_tau_i <- diag(1, k_w)
u_sigma_df_prior <- r # Prior degrees of freedom
u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix
u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom
# Initial values
beta <- matrix(c(1, -4), k_w, r)
u_sigma_i <- diag(.0001, k)
u_sigma <- solve(u_sigma_i)
g_i <- u_sigma_i
# Data containers
draws_alpha <- matrix(NA, k_alpha, store)
draws_beta <- matrix(NA, k_beta, store)
draws_pi <- matrix(NA, k * k_w, store)
draws_gamma <- matrix(NA, k_gamma, store)
draws_sigma <- matrix(NA, k^2, store)
# Start Gibbs sampler
for (draw in 1:iter) {
# Draw conditional mean parameters
temp <- post_coint_kls(y = y, beta = beta, w = w, x = x, sigma_i = u_sigma_i,
v_i = v_i, p_tau_i = p_tau_i, g_i = g_i,
gamma_mu_prior = a_mu_prior,
gamma_V_i_prior = a_v_i_prior)
alpha <- temp$alpha
beta <- temp$beta
Pi <- temp$Pi
gamma <- temp$Gamma
# Draw variance-covariance matrix
u <- y - Pi %*% w - matrix(gamma, k) %*% x
u_sigma_scale_post <- solve(tcrossprod(u) + v_i * alpha %*% tcrossprod(crossprod(beta, p_tau_i) %*% beta, alpha))
u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k)
u_sigma <- solve(u_sigma_i)
# Update g_i
g_i <- u_sigma_i
# Store draws
if (draw > burnin) {
draws_alpha[, draw - burnin] <- alpha
draws_beta[, draw - burnin] <- beta
draws_pi[, draw - burnin] <- Pi
draws_gamma[, draw - burnin] <- gamma
draws_sigma[, draw - burnin] <- u_sigma
}
}
Obtain point estimates as the mean of the parameter draws:
Gamma <- rowMeans(draws_gamma) # Obtain means for every row
Gamma <- matrix(Gamma, k) # Transform mean vector into a matrix
Gamma <- round(Gamma, 3) # Round values
dimnames(Gamma) <- list(dimnames(y)[[1]], dimnames(x)[[1]]) # Rename matrix dimensions
round(Gamma, 3) # Print
#> d.R.1 d.Dp.1 d.R.2 d.Dp.2 d.R.3 d.Dp.3 const season.1 season.2
#> d.R 0.262 -0.176 -0.021 -0.199 0.222 -0.096 0.002 0.001 0.009
#> d.Dp 0.079 -0.386 0.005 -0.422 0.024 -0.362 0.011 -0.034 -0.018
#> season.3
#> d.R 0.000
#> d.Dp -0.016
beta <- rowMeans(t(t(draws_beta) / t(draws_beta)[, 1])) # Obtain means for every row
beta <- matrix(beta, k_w) # Transform mean vector into a matrix
beta <- round(beta, 3) # Round values
dimnames(beta) <- list(dimnames(w)[[1]], NULL) # Rename matrix dimensions
beta # Print
#> [,1]
#> l.R 1.000
#> l.Dp -8.894
Sigma <- rowMeans(draws_sigma) # Obtain means for every row
Sigma <- matrix(Sigma, k) # Transform mean vector into a matrix
Sigma <- round(Sigma * 10^4, 2) # Round values
dimnames(Sigma) <- list(dimnames(y)[[1]], dimnames(y)[[1]]) # Rename matrix dimensions
Sigma # Print
#> d.R d.Dp
#> d.R 0.30 -0.02
#> d.Dp -0.02 0.27
bvec
objectsThe bvec
function can be used to collect output of the Gibbs sampler in a standardised object, which can be used further for forecasting, impulse response analysis or forecast error variance decomposition.
# Number of non-deterministic coefficients
k_nondet <- (k_x - 4) * k
# Generate bvec object
bvec_est <- bvec(y = y, w = w, x = x,
Pi = draws_pi,
Gamma = draws_gamma[1:k_nondet,],
C = draws_gamma[(k_nondet + 1):nrow(draws_gamma),],
Sigma = draws_sigma)
Posterior draws can be thinned with function thin
:
bvec_est <- thin(bvec_est, thin = 5)
The function bvec_to_bvar
can be used to transform the VEC model into a VAR in levels:
bvar_form <- bvec_to_bvar(bvec_est)
bvar_pred <- predict(bvar_form, n.ahead = 10, new_D = t(bvar_form$x[9:12, 91:100]))
plot(bvar_pred)
Impulse responses for VECs can be constructed from their VAR respresentations.
IR <- irf(bvar_form, impulse = "R", response = "Dp", n.ahead = 20)
plot(IR, main = "Forecast Error Impulse Response", xlab = "Year", ylab = "Response")
OIR <- irf(bvar_form, impulse = "R", response = "Dp", n.ahead = 20, type = "oir")
plot(OIR, main = "Orthogonalised Impulse Response", xlab = "Year", ylab = "Response")
GIR <- irf(bvar_form, impulse = "R", response = "Dp", n.ahead = 20, type = "gir")
plot(GIR, main = "Generalised Impulse Response", xlab = "Year", ylab = "Response")
bvec_fevd <- fevd(bvar_form, response = "Dp", n.ahead = 20)
plot(bvec_fevd, main = "FEVD of inflation")
Koop, G., León-González, R., & Strachan R. W. (2010). Efficient posterior simulation for cointegrated models with priors on the cointegration space. Econometric Reviews, 29(2), 224-242. https://doi.org/10.1080/07474930903382208
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. https://doi.org/10.1016/S0165-1765(97)00214-0