Posterior Draw from a Normal Distribution
Produces a draw of coefficients from a normal posterior density.
post_normal(y, x, sigma_i, a_prior, v_i_prior)
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. |
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 Σ.
For a given prior mean vector \underline{a} and prior covariance matrix \underline{V} the posterior covariance matrix is obtained by
\overline{V} = ≤ft[ \underline{V}^{-1} + ≤ft(X X^{\prime} \otimes Σ^{-1} \right) \right]^{-1}
and the posterior mean by
\overline{a} = \overline{V} ≤ft[ \underline{V}^{-1} \underline{a} + vec(Σ^{-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.
A vector.
Lütkepohl, H. (2006). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.
# 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.