bvartools (version 0.2.4)

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)

Value

A vector.

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.

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

Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

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)
x <- t(temp$data$Z)
k <- nrow(y)
tt <- 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) / tt)

# 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)