Bayesian Analysis

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

Jesse Windle and Carlos M. Carvalho

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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 Rn 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 Rn 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.

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

First available in Project Euclid: 21 November 2014

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backward sample forward filter realized covariance stochastic volatility


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.

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See also

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  • Related item: Catherine Scipione Forbes, Comment on Article by Windle and Carvalho. Bayesian Anal., Vol. 9, Iss. 4 (2014) 805–808.
  • Related item: Enrique ter Horst, German Molina, Comment on Article by Windle and Carvalho. Bayesian Anal., Vol. 9, Iss. 4 (2014) 809–819.
  • Related item: Jesse Windle, Carlos M. Carvalho (2014). Rejoinder. Bayesian Anal., Vol. 9, Iss. 4 (2014) 819–822.