post_coint_kls {bvartools} | R Documentation |
Posterior Draw for Cointegration Models
Description
Produces a draw of coefficients for cointegration models 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(
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 |
beta |
a |
w |
a |
sigma_i |
an inverse of the |
v_i |
a numeric between 0 and 1 specifying the shrinkage of the cointegration space prior. |
p_tau_i |
an inverted |
g_i |
a |
x |
a |
gamma_mu_prior |
a |
gamma_v_i_prior |
an inverted |
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 |
beta |
a draw of the |
Pi |
a draw of the |
Gamma |
a draw of the |
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. doi:10.1080/07474930903382208
Examples
# Load data
data("e6")
# Generate model data
temp <- gen_vec(e6, p = 1, r = 1)
y <- t(temp$data$Y)
ect <- t(temp$data$W)
k <- nrow(y) # Endogenous variables
tt <- ncol(y) # Number of observations
# Initial value of Sigma
sigma <- tcrossprod(y) / tt
sigma_i <- solve(sigma)
# Initial values of beta
beta <- matrix(c(1, -4), k)
# Draw parameters
coint <- post_coint_kls(y = y, beta = beta, w = ect, sigma_i = sigma_i,
v_i = 0, p_tau_i = diag(1, k), g_i = sigma_i)