post_coint_kls: Posterior Draw for Cointegration Models

In bvartools: Bayesian Inference of Vector Autoregressive and Error Correction Models

Posterior Draw for Cointegration Models

Description

Produces a draw of coefficients for cointegration models with a prior on the cointegration space as proposed in Koop et al. (2010) and a draw of non-cointegration coefficients from a normal density.

Usage

post_coint_kls(
  y,
  beta,
  w,
  sigma_i,
  v_i,
  p_tau_i,
  g_i,
  x = NULL,
  gamma_mu_prior = NULL,
  gamma_v_i_prior = NULL
)

Arguments

y	a K \times T matrix of differenced endogenous variables.
beta	a M \times r cointegration matrix \beta.
w	a M \times T matrix of variables in the cointegration term.
sigma_i	an inverse of the K \times K variance-covariance matrix.
v_i	a numeric between 0 and 1 specifying the shrinkage of the cointegration space prior.
p_tau_i	an inverted M \times M matrix specifying the central location of the cointegration space prior of sp(\beta).
g_i	a K \times K matrix.
x	a N \times T matrix of differenced regressors and unrestricted deterministic terms.
gamma_mu_prior	a KN \times 1 prior mean vector of non-cointegration coefficients.
gamma_v_i_prior	an inverted KN \times KN prior covariance matrix of non-cointegration coefficients.

Details

The function produces posterior draws of the coefficient matrices \alpha, \beta and \Gamma for the model

y_{t} = \alpha \beta^{\prime} w_{t-1} + \Gamma z_{t} + u_{t},

where y_{t} is a K-dimensional vector of differenced endogenous variables. w_{t} is an M \times 1 vector of variables in the cointegration term, which include lagged values of endogenous and exogenous variables in levels and restricted deterministic terms. z_{t} is an N-dimensional vector of differenced endogenous and exogenous explanatory variabes as well as unrestricted deterministic terms. The error term is u_t \sim \Sigma.

Draws of the loading matrix \alpha are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix is

\overline{V}_{\alpha} = \left[\left(v^{-1} (\beta^{\prime} P_{\tau}^{-1} \beta) \otimes G_{-1}\right) + \left(ZZ^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\alpha} = \overline{V}_{\alpha} + vec(\Sigma^{-1} Y Z^{\prime}),

where Y is a K \times T matrix of differenced endogenous variables and Z = \beta^{\prime} W with W as an M \times T matrix of variables in the cointegration term.

For a given prior mean vector \underline{\Gamma} and prior covariance matrix \underline{V_{\Gamma}} the posterior covariance matrix of non-cointegration coefficients in \Gamma is obtained by

\overline{V}_{\Gamma} = \left[ \underline{V}_{\Gamma}^{-1} + \left(X X^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{\Gamma} = \overline{V}_{\Gamma} \left[ \underline{V}_{\Gamma}^{-1} \underline{\Gamma} + vec(\Sigma^{-1} Y X^{\prime}) \right],

where X is an M \times T matrix of explanatory variables, which do not enter the cointegration term.

Draws of the cointegration matrix \beta are obtained using the prior on the cointegration space as proposed in Koop et al. (2010). The posterior covariance matrix of the unrestricted cointegration matrix B is

\overline{V}_{B} = \left[\left(A^{\prime} G^{-1} A \otimes v^{-1} P_{\tau}^{-1} \right) + \left(A^{\prime} \Sigma^{-1} A \otimes WW^{\prime} \right) \right]^{-1}

and the posterior mean by

\overline{B} = \overline{V}_{B} + vec(W Y_{B}^{-1} \Sigma^{-1} A),

where Y_{B} = Y - \Gamma X and A = \alpha (\alpha^{\prime} \alpha)^{-\frac{1}{2}}.

The final draws of \alpha and \beta are calculated using \beta = B (B^{\prime} B)^{-\frac{1}{2}} and \alpha = A (B^{\prime} B)^{\frac{1}{2}}.

Value

A named list containing the following elements:

alpha	a draw of the K \times r loading matrix.
beta	a draw of the M \times r cointegration matrix.
Pi	a draw of the K \times M cointegration matrix \Pi = \alpha \beta^{\prime}.
Gamma	a draw of the K \times N coefficient matrix for non-cointegration parameters.

References

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/07474930903382208")}

Examples

# Load data
data("e6")

# Generate model data
temp <- gen_vec(e6, p = 1, r = 1)
y <- t(temp$data$Y)
ect <- t(temp$data$W)

k <- nrow(y) # Endogenous variables
tt <- ncol(y) # Number of observations

# Initial value of Sigma
sigma <- tcrossprod(y) / tt
sigma_i <- solve(sigma)

# Initial values of beta
beta <- matrix(c(1, -4), k)

# Draw parameters
coint <- post_coint_kls(y = y, beta = beta, w = ect, sigma_i = sigma_i,
                        v_i = 0, p_tau_i = diag(1, k), g_i = sigma_i)

bvartools documentation built on May 29, 2024, 5:32 a.m.