Bayesian semiparametric multivariate stochastic volatility with application

Abstract

In this article, we establish a Cholesky-type multivariate stochastic volatility estimation framework, in which we let the innovation vector follow a Dirichlet process mixture (DPM), thus enabling us to model highly flexible return distributions. The Cholesky decomposition allows parallel univariate process modeling and creates potential for estimating high-dimensional specifications. We use Markov chain Monte Carlo methods for posterior simulation and predictive density computation. We apply our framework to a five-dimensional stock-return data set and analyze international stock-market co-movements among the largest stock markets. The empirical results show that our DPM modeling of the innovation vector yields substantial gains in out-of-sample density forecast accuracy when compared with the prevalent benchmark models.

Keywords:

• Bayesian nonparametrics
• Dirichlet process mixture
• Markov chain Monte Carlo
• stock-market co-movements

JEL CLASSIFICATION:

• C11
• C14
• C53
• C58
• G10

Acknowledgments

We are grateful to Esfandiar Maasoumi, an anonymous associate editor, and three reviewers for their constructive and extensive comments, which greatly improved the paper. The usual disclaimer applies.

Notes

1 We specify the $\text{AR}(1)$ process for ${\mathbf{h}}_t$ in Eq. (8)(8) $${\mathbf{h}}_t={\mathbf{\Phi }}_{\mathbf{h}}{\mathbf{h}}_{t-1}+{\mathit{\eta }}_t,$$(8) without an intercept term. This is due to an identification problem that would arise in the case of a nonzero intercept; see Jensen and Maheu (2010).

2 In the hierarchical representation, $\stackrel{\mathrm{d}}{=}$ means “has the distribution.” The operator $\text{diag}({\lambda }_1,\dots ,{\lambda }_m)$ creates the diagonal m × m matrix, say M, with ${\mathbf{M}}_{ii}={\lambda }_i$ and ${\mathbf{M}}_{ij}=0$ for $i\ne j$ $(i,j=1,\dots ,m).$

3 Prima facie, the diagonal structure of ${\mathbf{\Lambda }}_t$ might appear restrictive. However, as will become evident below, it does not impose any severe restriction on model flexibility.

4 In an ideal setting, we would set $B^{\max }$ equal to the true number of components in the data-generating process. Since this number is unknown in our empirical setup, we experimented with several $B^{\max }$ values and found that $B^{\max }=3$ produces accurate results.

5 Ausín et al. (2014) provide a detailed discussion on the appropriate choice of prior distributions. We use the same prior distributions in our empirical application in Section 4.

6 We thank two reviewers for drawing our attention to this issue.

7 Since the data turn out not to be very informative about the hyperparameters c_i, we also experimented with other priors for c_i. While the posterior distributions of the hyperparameters c_i are affected, the posterior distributions of the other model parameters do not change substantially.

8 A sensitivity analysis for the parameter c_i (or ${\tilde{c}}_i$) reveals that the shape of its posterior is strongly affected by the choice of the prior. This finding is, however, inconsequential, as different specifications of the prior for c_i have only minor impact on the posterior distributions of all (but one) remaining model parameters. The only parameter, affected by the prior of c_i, is the average number of nonempty clusters. This illustrates one of the most prominent features of the Bayesian nonparametric models, namely that the same density can be approximated using different numbers of clusters with different mixing parameters.

9 To economize on space, our choice of benchmark models is limited to these three specifications. Another model, not considered here, is the MGARCH-DPM model (Jensen and Maheu, 2013). Jensen and Maheu (2013) propose a nonstochastic (GARCH-type) approach to multivariate volatility modeling, which (i) is order invariant, and (ii) allows for nondiagonal mixing scale. A comparison of their GARCH-type model with our stochastic volatility approach will be subject to future research.

10 It is well-known that the sampler of Jacquier et al. (2002) has some inefficiencies that slow down mixing. Chib et al. (2002) and Jensen and Maheu (2010) propose more efficient sampling algorithms. Chib et al. (2002) overcame the naturally built-up dependency between the parameters and the latent volatilities. Jensen and Maheu (2010) suggested a random-blocking approach so that the dependency on the beginning and ending volatilities are mixed over. However, due to the nonparametric part of our model, these algorithms cannot easily be adopted. Since our trace plots do not indicate poor mixing, we propose to use the sampler of Jacquier et al. (2002).