bvartools (version 0.2.4)

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, svd = FALSE)

Value

A vector.

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.

svd

logical. If TRUE the singular value decomposition is used to determine the root of the posterior covariance matrix. Default is FALSE which means that the eigenvalue decomposition is used.

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

# Load data
data("e1")
data <- diff(log(e1))

# Generate model data
temp <- gen_var(data, p = 2, deterministic = "const")
y <- t(temp$data$Y)
z <- temp$data$SUR
k <- nrow(y)
tt <- ncol(y)
m <- ncol(z)

# 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) / tt)

# 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)