## Bayesian Analysis

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

#### 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 $\mathbb{R}^{n}$ 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 $\mathbb{R}^{n}$ 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.

#### Article information

Source
Bayesian Anal., Volume 9, Number 4 (2014), 759-792.

Dates
First available in Project Euclid: 21 November 2014

https://projecteuclid.org/euclid.ba/1416579176

Digital Object Identifier
doi:10.1214/14-BA888

Mathematical Reviews number (MathSciNet)
MR3293953

Zentralblatt MATH identifier
1327.62170

#### Citation

Windle, Jesse; Carvalho, Carlos M. A Tractable State-Space Model for Symmetric Positive-Definite Matrices. Bayesian Anal. 9 (2014), no. 4, 759--792. doi:10.1214/14-BA888. https://projecteuclid.org/euclid.ba/1416579176

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