bvartools (version 0.0.1)

post_coint_kls_sur: Posterior Draw for Cointegration Models

Description

Produces a draw of coefficients for cointegration models in SUR form 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_sur(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

the inverse of the constant $K\times K$ error variance-covariance matrix. For time varying variance-covariance matrics a $KT\times K$ can be provided.

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 $K\times K$ or $KT\times K$ matrix.

x

a $KT\times NK$ 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.

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 $\mathrm{\Pi }=\alpha {\beta }^{\prime }$.

Gamma

a draw of the $K\times N$ coefficient matrix for non-cointegration parameters.

Details

The function produces posterior draws of the coefficient matrices $\alpha$, $\beta$ and $\mathrm{\Gamma }$ for the model $$y_t=\alpha {\beta }^{\prime }{w}_{t-1}+\mathrm{\Gamma }{z}_t+u_t,$$ 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 $u_t\sim \mathrm{\Sigma }$.

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 $${\bar{V}}_{\alpha }={[(v^{-1}({\beta }^{\prime }{P}_{\tau }^{-1}\beta )\otimes G_{-1})+(Z{Z}^{\prime }\otimes {\mathrm{\Sigma }}^{-1})]}^{-1}$$ and the posterior mean by $$\bar{\alpha }={\bar{V}}_{\alpha }+vec({\mathrm{\Sigma }}^{-1}Y{Z}^{\prime }),$$ 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 $\underset{―}{\mathrm{\Gamma }}$ and prior covariance matrix $\underset{―}{V_{\mathrm{\Gamma }}}$ the posterior covariance matrix of non-cointegration coefficients in $\mathrm{\Gamma }$ is obtained by $${\bar{V}}_{\mathrm{\Gamma }}={[{\underset{―}{V}}_{\mathrm{\Gamma }}^{-1}+(X{X}^{\prime }\otimes {\mathrm{\Sigma }}^{-1})]}^{-1}$$ and the posterior mean by $$\bar{\mathrm{\Gamma }}={\bar{V}}_{\mathrm{\Gamma }}[{\underset{―}{V}}_{\mathrm{\Gamma }}^{-1}\underset{―}{\mathrm{\Gamma }}+vec({\mathrm{\Sigma }}^{-1}Y{X}^{\prime })],$$ 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 $${\bar{V}}_B={[(A^{\prime }{G}^{-1}A\otimes v^{-1}{P}_{\tau }^{-1})+(A^{\prime }{\mathrm{\Sigma }}^{-1}A\otimes W{W}^{\prime })]}^{-1}$$ and the posterior mean by $$\bar{B}={\bar{V}}_B+vec(W{Y}_B^{-1}{\mathrm{\Sigma }}^{-1}A),$$ where $Y_B=Y-\mathrm{\Gamma }X$ 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 $\alpha =A(B^{\prime }B{)}^{\frac{1}{2}}$.

References

Koop, G., Le<U+00F3>n-Gonz<U+00E1>lez, 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

Examples

# NOT RUN {
data("e6")
temp <- gen_vec(e6, p = 0)
y <- temp$Y
ect <- temp$W

k <- nrow(y)
t <- ncol(y)

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

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

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

# }