post_coint_kls_sur

Posterior Draw for Cointegration Models

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)

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 \(\alpha = A (B^{\prime} B)^{\frac{1}{2}}\).

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

Aliases

• post_coint_kls_sur

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)

# }