bvartools (version 0.0.1)

post_normal: Posterior Draw from a Normal Distribution

Description

Produces a draw of coefficients from a normal posterior density.

Usage

post_normal(y, x, sigma_i, a_prior, v_i_prior)

Arguments

y

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

x

an \(M \times T\) matrix of explanatory variables.

sigma_i

the inverse of the \(K \times K\) variance-covariance matrix.

a_prior

a \(KM \times 1\) numeric vector of prior means.

v_i_prior

the inverse of the \(KM \times KM\) prior covariance matrix.

Value

A vector.

Details

The function produces a vectorised posterior draw \(a\) of the \(K \times M\) coefficient matrix \(A\) for the model $$y_{t} = A x_{t} + u_{t},$$ where \(y_{t}\) is a K-dimensional vector of endogenous variables, \(x_{t}\) is an M-dimensional vector of explanatory variabes and the error term is \(u_t \sim \Sigma\).

For a given prior mean vector \(\underline{a}\) and prior covariance matrix \(\underline{V}\) the posterior covariance matrix is obtained by $$\overline{V} = \left[ \underline{V}^{-1} + \left(X X^{\prime} \otimes \Sigma^{-1} \right) \right]^{-1}$$ and the posterior mean by $$\overline{a} = \overline{V} \left[ \underline{V}^{-1} \underline{a} + vec(\Sigma^{-1} Y X^{\prime}) \right],$$ where \(Y\) is a \(K \times T\) matrix of the endogenous variables and \(X\) is an \(M \times T\) matrix of the explanatory variables.

References

L<U+00FC>tkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

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)
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(y = y, x = x, sigma_i = sigma_i,
                 a_prior = a_mu_prior, v_i_prior = a_v_i_prior)

# }