post_coint_kls_sur

0th

Percentile

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)
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.

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., 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)

# }
Documentation reproduced from package bvartools, version 0.0.1, License: GPL (>= 2)

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