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

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Windle, Jesse
Carvalho, Carlos M.

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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 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 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.


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Windle, Jesse, and Carlos M. Carvalho. "A tractable state-space model for symmetric positive-definite matrices." Bayesian Analysis, Vol. 9, No. 4 (Dec., 2014): 759-792.