A Multivariate Graphical Stochastic Volatility Model

TL;DR

This paper reviews recent advances in Gaussian Graphical Models (GGM) and introduces a new efficient method for GGM comparison, culminating in a hierarchical multivariate stochastic volatility model that adapts well to market volatility changes.

Contribution

It develops a computationally efficient algorithm for GGM comparison and integrates GGM uncertainty into a hierarchical stochastic volatility model for financial data.

Findings

The new algorithm significantly reduces computation time.

The hierarchical model effectively captures market volatility swings.

The model improves posterior predictive calibration during market crises.

Abstract

The Gaussian Graphical Model (GGM) is a popular tool for incorporating sparsity into joint multivariate distributions. The G-Wishart distribution, a conjugate prior for precision matrices satisfying general GGM constraints, has now been in existence for over a decade. However, due to the lack of a direct sampler, its use has been limited in hierarchical Bayesian contexts, relegating mixing over the class of GGMs mostly to situations involving standard Gaussian likelihoods. Recent work, however, has developed methods that couple model and parameter moves, first through reversible jump methods and later by direct evaluation of conditional Bayes factors and subsequent resampling. Further, methods for avoiding prior normalizing constant calculations--a serious bottleneck and source of numerical instability--have been proposed. We review and clarify these developments and then propose a new methodology for GGM comparison that blends many recent themes. Theoretical developments and computational timing experiments reveal an algorithm that has limited computational demands and dramatically improves on computing times of existing methods. We conclude by developing a parsimonious multivariate stochastic volatility model that embeds GGM uncertainty in a larger hierarchical framework. The method is shown to be capable of adapting to the extreme swings in market volatility experienced in 2008 after the collapse of Lehman Brothers, offering considerable improvement in posterior predictive distribution calibration.