bvec function

Bayesian Vector Error Correction Objects

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

A plot function for objects of class "bvec".

bvec(
  y,
  alpha = NULL,
  beta = NULL,
  beta_x = NULL,
  beta_d = 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
)

## S3 method for class 'bvec'
plot(x, ci = 0.95, type = "hist", ...)

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 $Kr \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$

  corresponding to the endogenous variables of the model.

• beta_x: a $Mr \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$

  corresponding to unmodelled, non-deterministic variables.

• beta_d: a $N^{R}r \times S$ matrix of MCMC coefficient draws of cointegration matrix $\beta$

  corresponding to restricted deterministic terms.

• 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: an object of class "bvec", usually, a result of a call to draw_posterior.

• 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.

• ci: interval used to calculate credible bands for time-varying parameters.

• type: either "hist" (default) for histograms, "trace" for a trace plot or "boxplot" for a boxplot. Only used for parameter draws of constant coefficients.

• ...: further graphical parameters.

Returns

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. In case of time varying parameters a list of such objects.

• A0_lambda: an $S \times K^2$ "mcmc" object of inclusion parameters for coefficients corresponding to structural parameters.

• A0_sigma: an $S \times K^2$ "mcmc" object of the error covariance matrices of the structural parameters in a model with time varying parameters.

• alpha: an $S \times Kr$ "mcmc" object of coefficient draws of loading parameters. In case of time varying parameters a list of such objects.

• beta: an $S \times ((K + M + N^{R})r)$ "mcmc" object of coefficient draws of cointegration parameters corresponding to the endogenous variables of the model. In case of time varying parameters a list of such objects.

• beta_x: an $S \times KM$ "mcmc" object of coefficient draws of cointegration parameters corresponding to unmodelled, non-deterministic variables. In case of time varying parameters a list of such objects.

• beta_d: an $S \times KN^{R}$ "mcmc" object of coefficient draws of cointegration parameters corresponding to restricted deterministic variables. In case of time varying parameters a list of such objects.

• Pi: an $S \times K^2$ "mcmc" object of coefficient draws of endogenous variables in the cointegration matrix. In case of time varying parameters a list of such objects.

• Pi_x: an $S \times KM$ "mcmc" object of coefficient draws of unmodelled, non-deterministic variables in the cointegration matrix. In case of time varying parameters a list of such objects.

• Pi_d: an $S \times KN^{R}$ "mcmc" object of coefficient draws of restricted deterministic variables in the cointegration matrix. In case of time varying parameters a list of such objects.

• Gamma: an $S \times (p-1)K^2$ "mcmc" object of coefficient draws of differenced lagged endogenous variables. In case of time varying parameters a list of such objects.

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

• Gamma_sigma: an $S \times (p - 1)K^2$ "mcmc" object of the error covariance matrices of the coefficients of lagged endogenous variables in a model with time varying parameters.

• Upsilon: an $S \times sMK$ "mcmc" object of coefficient draws of differenced unmodelled, non-deterministic variables. In case of time varying parameters a list of such objects.

• Upsilon_lambda: an $S \times sMK$ "mcmc" object of inclusion parameters for coefficients corresponding to differenced unmodelled, non-deterministic variables.

• Upsilon_sigma: an $S \times sMK$ "mcmc" object of the error covariance matrices of the coefficients of unmodelled, non-deterministic variables in a model with time varying parameters.

• C: an $S \times KN^{UR}$ "mcmc" object of coefficient draws of deterministic terms that do not enter the cointegration term. In case of time varying parameters a list of such objects.

• C_lambda: an $S \times KN^{UR}$ "mcmc" object of inclusion parameters for coefficients corresponding to deterministic terms, that do not enter the conintegration term.

• C_sigma: an $S \times KN^{UR}$ "mcmc" object of the error covariance matrices of the coefficients of deterministic terms, which do not enter the cointegration term, in a model with time varying parameters.

• Sigma: an $S \times K^2$ "mcmc" object of variance-covariance draws. In case of time varying parameters a list of such objects.

• Sigma_lambda: an $S \times K^2$ "mcmc" object inclusion parameters for the variance-covariance matrix.

• Sigma_sigma: an $S \times K^2$ "mcmc" object of the error covariance matrices of the coefficients of the error covariance matrix of the measurement equation of a model with time varying parameters.

• specifications: a list containing information on the model specification.

Details

For the vector error correction model with unmodelled exogenous variables (VECX)

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 $u_t \sim N(0, \Sigma_u)$.

For time varying parameter and stochastic volatility models the respective coefficients and error covariance matrix of the above model are assumed to be time varying, respectively.

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

# 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)

# Load data 
data("e6")

# Generate model
model <- gen_vec(data = e6, p = 2, r = 1, const = "unrestricted",
                 iterations = 20, burnin = 10)
# Chosen number of iterations and burn-in should be much higher.

# Add priors
model <- add_priors(model)

# Obtain posterior draws
object <- draw_posterior(model)

# Plot draws
plot(object)