A Tractable State-Space Model for Symmetric Positive-Definite Matrices

• Activity Number: 4
• Type: Invited
• Date/Time: Sunday, August 9, 2015 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Jesse Windle* and Carlos M. Carvalho
• Companies: Duke University and The University of Texas at Austin
• Keywords: backward sample ; forward filter ; realized covariance ; stochastic volatility

Abstract

The Bayesian analysis of a state-space model includes computing the posterior distribution of the system's parameters as well as its latent states. When the latent states wander around Euclidean space there are several well-known modeling components and computational tools that may be profitably combined to achieve this task. When the latent states are constrained to a strict subset of Euclidean space these models and tools are either impaired or break down completely. State-space models whose latent states are covariance matrices arise in finance and exemplify the challenge of devising tractable models in the constrained setting. To that end, we present a state-space model whose observations and latent states take values on the manifold of symmetric positive-definite matrices and for which one may easily compute the posterior distribution of the latent states and the system's parameters as well as filtered distributions and one-step ahead predictions. Employing the model within the context of finance, we show how one can use realized covariance matrices as data to predict latent time-varying covariance matrices. This approach out-performs factor stochastic volatility.