asv_logML {ASV}

Compute the logarithm of the marginal likelihood for the stochastic volatility models with leverage

Description

This function computes the logarithm of the marginal likelihood for stochastic volatility models with leverage (asymmetric stochastic volatility models):

Usage

asv_logML(H, Theta, Theta_star, Y, iI = NULL, iM = NULL, vHyper = NULL)

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, rho)

Theta_star

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

Y

T x 1 vector of returns

iI

the number of particles to approximate the filtering density. Default is 5000.

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, a_1, b_1, n_0, S_0). Defaults is (0,1000, 1, 1, 1, 1, 0.01, 0.01)

Value

4 x 2 matrix.

Column 1

The logarithms of the marginal likelihood, the likelihood, the prior density and the posterior density.

Column 2

The standard errors of the logarithms of the marginal likelihood, the likelihood, the prior density and the posterior density.

Author(s)

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; rho = -0.3;
h  = 0;   Y = c();
for(i in 1:nobs){
  eps = rnorm(1, 0, 1)
  eta = rho*sigma_eta*eps + sigma_eta*sqrt(1-rho^2)*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 = 300; nburn = 100;
vhyper = c(0.0,1000,1.0,1.0,1.0,1.0,0.01,0.01)
out  = asv_mcmc(Y, nsim, nburn, vhyper)
vmu = out[[1]]; vphi = out[[2]]; vsigma_eta = out[[3]]; vrho = out[[4]];mh  = out[[5]];
mu  = mean(vmu); phi = mean(vphi); sigma_eta = mean(vsigma_eta);
rho = mean(vrho);
#
h          = mh[nsim,]
theta      = c(vmu[nsim],vphi[nsim],vsigma_eta[nsim],vrho[nsim])
theta_star = c(mu, phi, sigma_eta, rho)

# Increase iM in practice (such as iI = 5000, iM =5000).
result = asv_logML(h, theta, theta_star, Y, 100, 100, vHyper = vhyper)
result1     = matrix(0, 4, 2)
result1[,1] =result[[1]]
result1[,2] =result[[2]]
  
colnames(result1) = c("Estimate", "Std Err")
rownames(result1) = c("Log marginal lik", "Log likelihood", "Log prior", "Log posterior")
print(result1, digit=4)