post_coint_kls_sur {bvartools}R Documentation

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 β.

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(β).

g_i

a K \times K or KT \times K matrix. If the matrix is KT \times K, the function will automatically produce a K \times K matrix containing the means of the time varying K \times K covariance 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 α, β and Γ for the model

y_{t} = α β^{\prime} w_{t-1} + Γ 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 Σ.

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

\overline{V}_{α} = ≤ft[≤ft(v^{-1} (β^{\prime} P_{τ}^{-1} β) \otimes G_{-1}\right) + ≤ft(ZZ^{\prime} \otimes Σ^{-1} \right) \right]^{-1}

and the posterior mean by

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

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

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

\overline{V}_{Γ} = ≤ft[ \underline{V}_{Γ}^{-1} + ≤ft(X X^{\prime} \otimes Σ^{-1} \right) \right]^{-1}

and the posterior mean by

\overline{Γ} = \overline{V}_{Γ} ≤ft[ \underline{V}_{Γ}^{-1} \underline{Γ} + vec(Σ^{-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 β 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} = ≤ft[≤ft(A^{\prime} G^{-1} A \otimes v^{-1} P_{τ}^{-1} \right) + ≤ft(A^{\prime} Σ^{-1} A \otimes WW^{\prime} \right) \right]^{-1}

and the posterior mean by

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

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

The final draws of α and β are calculated using β = B (B^{\prime} B)^{-\frac{1}{2}} and α = 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 Π = α β^{\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. https://doi.org/10.1080/07474930903382208

Examples

data("e6")
temp <- gen_vec(e6, p = 1)
y <- temp$Y
ect <- temp$W

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

# 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, m),
                            g_i = sigma_i)


[Package bvartools version 0.0.2 Index]