# 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)
```

##### 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:

a draw of the \(K \times r\) loading matrix.

a draw of the \(M \times r\) cointegration matrix.

a draw of the \(K \times M\) cointegration matrix \(\Pi = \alpha \beta^{\prime}\).

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

##### 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)*