2
$\begingroup$

I want to understand the survey of Lopes and Polson (2010) regarding MV stochastic volatility. Assume the $p$-dimensional vector $y_t$ follows

$$y_t\sim N(\Theta,\Sigma_t).$$

In order to model the dynamic behavior of the covariance matrix one popular approach is to impose a factor analysis structure of the form

$$\Sigma_t=\beta H_t\beta'+\Psi_t$$

with $\beta$ the $p\times k$ matrix of factor loadings which is lower block triangular with diagonal elements equal to one. $\Psi_t$ and $H_t$ are diagonal covariance matrices.

On page 15 they point out that a large class of factor stochastic volatility models is written as

$$y_t|f_t,\beta,\Sigma_t \sim N(\Theta+\beta f_t;\Psi_t)\\f_t |H_t \sim N(0;H_t)$$

The elements of $\Psi_t$ are modeled by cond. independend univariate SV structures, while $\log h_t=(\log h_{1t},\ldots,\log h_{kt})'$ follows a first-order vector autoregression:

$$\log h_t |h_{t-1},\beta_0,\beta_1,U \sim N(\beta_0+\beta_1 \log h_{t-1},U)$$.

Let's assume conditionally independent components of the priors:

$$p(\theta)p(X)p(\Psi)p(\beta_0)p(\beta_1)p(U).$$

I want to find a way to sample from such a model via MCMC, however, many links are missing. Is there a standard implementation available for that or any paper describing the sampler in some detail? I am happy for each and every comment regarding this project.

$\endgroup$
2
$\begingroup$

This book chapter is a pretty good explanation of MCMC techniques for general state space models; you might want to start with this.

For your particular model here, instead of looking at a review paper, you can have a look at this paper, which features a rather involved Metropolis-within-Blocked-Gibbs sampling procedure. It is all detailed in Appendix B. Many of the distributions are multivariate Gaussian, and for those that aren't, they either use a Metropolis-Hastings sampler, or a method discussed in this other paper. This is the other paper that I know of that implements this model, and they use a different Gibbs technique.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.