ASV (version 1.1.4)

sv_posterior: Compute the logarithm of the posterior density for the stochastic volatility models without leverage

Description

This function computes the logarithm of the posterior density for stochastic volatility models without leverage (symmetric stochastic volatility models):

Usage

sv_posterior(H, Theta, Theta_star, Y, iM = NULL, vHyper = NULL)

Value

2 x 1 vector. The first element is the logarithm of the posterior density, and the second element is its standard error.

Arguments

H

T x 1 vector of latent log volatilities to start the reduced MCMC run to compute the log posterior density.

Theta

a vector of parameters to start the reduced MCMC run to compute the log posterior density. Theta = c(mu, phi, sigma_eta)

Theta_star

a vector of parameters to evaluate the log posterior density. Theta_star = c(mu, phi, sigma_eta)

Y

T x 1 vector of returns

iM

the number of iterations for the reduced MCMC run. Default is 5000.

vHyper

a vector of hyper-parameters to evaluate the log posterior density. vHyper = c(mu_0, sigma_0, a_0, b_0, n_0, S_0). Defaults is (0,1000, 1, 1, 0.01, 0.01)

Author

Yasuhiro Omori

References

Chib, S., and Jeliazkov, I. (2001). Marginal likelihood from the Metropolis-Hastings output. Journal of the American statistical association, 96(453), 270-281.

Examples

set.seed(111)
nobs = 80; # n is often larger than 1000 in practice.
mu = 0; phi = 0.97; sigma_eta = 0.3;
h  = 0;   Y = c();
for(i in 1:nobs){
  eps = rnorm(1, 0, 1)
  eta = sigma_eta*rnorm(1, 0, 1)
  y   = eps * exp(0.5*h)
  h   = mu + phi * (h-mu) + eta
  Y   = append(Y, y)
}

# This is a toy example. Increase nsim and nburn
# until the convergence of MCMC in practice.
nsim = 500; nburn = 100;
vhyper = c(0.0,1000,1.0,1.0,0.01,0.01)
out  = sv_mcmc(Y, nsim, nburn, vhyper)
vmu = out[[1]]; vphi = out[[2]]; vsigma_eta = out[[3]]; mh  = out[[4]];
mu  = mean(vmu); phi = mean(vphi); sigma_eta = mean(vsigma_eta);
#
h          = mh[nsim,]
theta      = c(vmu[nsim],vphi[nsim],vsigma_eta[nsim])
theta_star = c(mu, phi, sigma_eta)

# Increase iM in practice (such as iM =5000).
sv_posterior(h, theta, theta_star, Y, 100, vhyper)