Posterior Draw for Cointegration Models
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.
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 )
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 |
an inverse of the K \times K variance-covariance matrix. |
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 matrix. |
x |
a N \times T 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. |
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}}.
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. |
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
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.