bvartools (version 0.0.1)

post_normal_sur: Posterior Draw from a Normal Distribution

Description

Produces a draw of coefficients from a normal posterior density for a model with seemingly unrelated regresssions (SUR).

Usage

post_normal_sur(y, z, sigma_i, a_prior, v_i_prior)

Arguments

y

a \(K \times T\) matrix of endogenous variables.

z

a \(KT \times M\) matrix of explanatory variables.

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.

a_prior

a \(M x 1\) numeric vector of prior means.

v_i_prior

the inverse of the \(M x M\) prior covariance matrix.

Value

A vector.

Details

The function produces a posterior draw of the coefficient vector \(a\) for the model $$y_{t} = Z_{t} a + u_{t},$$ where \(u_t \sim N(0, \Sigma_{t})\). \(y_t\) is a K-dimensional vector of endogenous variables and \(Z_t = z_t^{\prime} \otimes I_K\) is a \(K \times KM\) matrix of regressors with \(z_t\) as a vector of regressors.

For a given prior mean vector \(\underline{a}\) and prior covariance matrix \(\underline{V}\) the posterior covariance matrix is obtained by $$\overline{V} = \left[ \underline{V}^{-1} + \sum_{t=1}^{T} Z_{t}^{\prime} \Sigma_{t}^{-1} Z_{t} \right]^{-1}$$ and the posterior mean by $$\overline{a} = \overline{V} \left[ \underline{V}^{-1} \underline{a} + \sum_{t=1}^{T} Z_{t}^{\prime} \Sigma_{t}^{-1} y_{t} \right].$$

Examples

# NOT RUN {
# Prepare data
data("e1")
data <- diff(log(e1))
temp <- gen_var(data, p = 2, deterministic = "const")
y <- temp$Y
x <- temp$Z
k <- nrow(y)
z <- kronecker(t(x), diag(1, k))
t <- ncol(y)
m <- k * nrow(x)

# Priors
a_mu_prior <- matrix(0, m)
a_v_i_prior <- diag(0.1, m)

# Initial value of inverse Sigma
sigma_i <- solve(tcrossprod(y) / t)

# Draw parameters
a <- post_normal_sur(y = y, z = z, sigma_i = sigma_i,
                     a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

# }