bvartools (version 0.1.0)

bvec: Bayesian Vector Error Correction Objects

Description

`bvec` is used to create objects of class "bvec".

Usage

bvec(
  y,
  alpha = NULL,
  beta = NULL,
  r = NULL,
  Pi = NULL,
  Pi_x = NULL,
  Pi_d = NULL,
  w = NULL,
  w_x = NULL,
  w_d = NULL,
  Gamma = NULL,
  Upsilon = NULL,
  C = NULL,
  x = NULL,
  x_x = NULL,
  x_d = NULL,
  A0 = NULL,
  Sigma = NULL,
  data = NULL,
  exogen = NULL
)

Arguments

y

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

alpha

a \(Kr \times S\) matrix of MCMC coefficient draws of the loading matrix \(\alpha\).

beta

a \(((K + M + N^{R})r) \times S\) matrix of MCMC coefficient draws of cointegration matrix \(\beta\).

r

an integer of the rank of the cointegration matrix.

Pi

a \(K^2 \times S\) matrix of MCMC coefficient draws of endogenous varaibles in the cointegration matrix.

Pi_x

a \(KM \times S\) matrix of MCMC coefficient draws of unmodelled, non-deterministic variables in the cointegration matrix.

Pi_d

a \(KN^{R} \times S\) matrix of MCMC coefficient draws of restricted deterministic terms.

w

a time-series object of lagged endogenous variables in levels, which enter the cointegration term, usually, a result of a call to gen_vec.

w_x

a time-series object of lagged unmodelled, non-deterministic variables in levels, which enter the cointegration term, usually, a result of a call to gen_vec.

w_d

a time-series object of deterministic terms, which enter the cointegration term, usually, a result of a call to gen_vec.

Gamma

a \((p-1)K^2 \times S\) matrix of MCMC coefficient draws of differenced lagged endogenous variables or a named list, where element coeffs contains a \((p - 1)K^2 \times S\) matrix of MCMC coefficient draws of lagged differenced endogenous variables and element lambda contains the corresponding draws of inclusion parameters in case variable selection algorithms were employed.

Upsilon

an \(sMK \times S\) matrix of MCMC coefficient draws of differenced unmodelled, non-deterministic variables or a named list, where element coeffs contains a \(sMK \times S\) 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.

C

an \(KN^{UR} \times S\) matrix of MCMC coefficient draws of unrestricted deterministic terms or a named list, where element coeffs contains a \(KN^{UR} \times S\) 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.

x

a time-series object of \(K(p - 1)\) differenced endogenous variables.

x_x

a time-series object of \(Ms\) differenced unmodelled regressors.

x_d

a time-series object of \(N^{UR}\) deterministic terms that do not enter the cointegration term.

A0

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.

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.

data

the original time-series object of endogenous variables.

exogen

the original time-series object of unmodelled variables.

Value

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

data

the original time-series object of endogenous variables.

exogen

the original time-series object of unmodelled variables.

y

a time-series object of differenced endogenous variables.

w

a time-series object of lagged endogenous variables in levels, which enter the cointegration term.

w_x

a time-series object of lagged unmodelled, non-deterministic variables in levels, which enter the cointegration term.

w_d

a time-series object of deterministic terms, which enter the cointegration term.

x

a time-series object of \(K(p - 1)\) differenced endogenous variables

x_x

a time-series object of \(Ms\) differenced unmodelled regressors.

x_d

a time-series object of \(N^{UR}\) deterministic terms that do not enter the cointegration term.

A0

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 coefficients corresponding to structural parameters.

alpha

an \(S \times Kr\) "mcmc" object of coefficient draws of loading parameters.

beta

an \(S \times ((K + M + N^{R})r)\) "mcmc" object of coefficient draws of cointegration parameters.

Pi

an \(S \times K^2\) "mcmc" object of coefficient draws of endogenous variables in the cointegration matrix.

Pi_x

an \(S \times KM\) "mcmc" object of coefficient draws of unmodelled, non-deterministic variables in the cointegration matrix.

Pi_d

an \(S \times KN^{R}\) "mcmc" object of coefficient draws of unrestricted deterministic variables in the cointegration matrix.

Gamma

an \(S \times (p-1)K^2\) "mcmc" object of coefficient draws of differenced lagged endogenous variables.

Gamma_lamba

an \(S \times (p-1)K^2\) "mcmc" object of inclusion parameters for coefficients corresponding to differenced lagged endogenous variables.

Upsilon

an \(S \times sMK\) "mcmc" object of coefficient draws of differenced unmodelled variables.

Upsilon_lambda

an \(S \times sMK\) "mcmc" object of inclusion parameters for coefficients corresponding to differenced unmodelled, non-deterministic variables.

C

an \(S \times KN^{UR}\) "mcmc" object of coefficient draws of deterministic terms that do not enter the cointegration term.

C_lambda

an \(S \times KN^{UR}\) "mcmc" object of inclusion parameters for coefficients corresponding to deterministic terms, that do not enter the conitegration term.

Sigma

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

Sigma_lambda

an \(S \times K^2\) "mcmc" object inclusion parameters for the variance-covariance matrix.

specifications

a list containing information on the model specification.

Details

For the vector error correction model with unmodelled exogenous variables (VECX) $$A_0 \Delta y_t = \Pi^{+} \begin{pmatrix} y_{t-1} \\ x_{t-1} \\ d^{R}_{t-1} \end{pmatrix} + \sum_{i = 1}^{p-1} \Gamma_i \Delta y_{t-i} + \sum_{i = 0}^{s-1} \Upsilon_i \Delta x_{t-i} + C^{UR} d^{UR}_t + u_t$$ the function collects the \(S\) draws of a Gibbs sampler in a standardised object, where \(\Delta y_t\) is a K-dimensional vector of differenced endogenous variables and \(A_0\) is a \(K \times K\) matrix of structural coefficients. \(\Pi^{+} = \left[ \Pi, \Pi^{x}, \Pi^{d} \right]\) is the coefficient matrix of the error correction term, where \(y_{t-1}\), \(x_{t-1}\) and \(d^{R}_{t-1}\) are the first lags of endogenous, exogenous variables in levels and restricted deterministic terms, respectively. \(\Pi\), \(\Pi^{x}\), and \(\Pi^{d}\) are the corresponding coefficient matrices, respectively. \(\Gamma_i\) is a coefficient matrix of lagged differenced endogenous variabels. \(\Delta x_t\) is an M-dimensional vector of unmodelled, non-deterministic variables and \(\Upsilon_i\) its corresponding coefficient matrix. \(d_t\) is an \(N^{UR}\)-dimensional vector of unrestricted deterministics and \(C^{UR}\) the 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 alpha, beta, Pi, Pi_x, Pi_d, A0, Gamma, Ypsilon, C and Sigma have to correspond to the same MCMC iteration.

Examples

# NOT RUN {
# Load data
data("e6")
# Generate model
data <- gen_vec(e6, p = 4, r = 1, const = "unrestricted", season = "unrestricted")
# Obtain data matrices
y <- t(data$data$Y)
w <- t(data$data$W)
x <- t(data$data$X)

# Reset random number generator for reproducibility
set.seed(1234567)

iterations <- 400 # 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
draws <- iterations + burnin

r <- 1 # Set rank

tt <- 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 <- tt + u_sigma_df_prior # Posterior degrees of freedom

# Initial values
beta <- matrix(c(1, -4), k_w, r)
u_sigma_i <- diag(1 / .0001, k)
g_i <- u_sigma_i

# Data containers
draws_alpha <- matrix(NA, k_alpha, iterations)
draws_beta <- matrix(NA, k_beta, iterations)
draws_pi <- matrix(NA, k * k_w, iterations)
draws_gamma <- matrix(NA, k_gamma, iterations)
draws_sigma <- matrix(NA, k^2, iterations)

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

# Number of non-deterministic coefficients
k_nondet <- (k_x - 4) * k

# Generate bvec object
bvec_est <- bvec(y = data$data$Y, w = data$data$W,
                 x = data$data$X[, 1:6],
                 x_d = data$data$X[, 7:10],
                 Pi = draws_pi,
                 Gamma = draws_gamma[1:k_nondet,],
                 C = draws_gamma[(k_nondet + 1):nrow(draws_gamma),],
                 Sigma = draws_sigma)

# }