Bayesian Analysis

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

Jesse Windle and Carlos M. Carvalho

Full-text: Open access

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 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.

Article information

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

Dates
First available in Project Euclid: 21 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.ba/1416579176

Digital Object Identifier
doi:10.1214/14-BA888

Mathematical Reviews number (MathSciNet)
MR3293953

Zentralblatt MATH identifier
1327.62170

Keywords
backward sample forward filter realized covariance stochastic volatility

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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References

  • Aguilar, O. and West, M. (2000). “Bayesian Dynamic Factor Models and Portfolio Allocation.” Journal of Business and Economic Statistics, 18(3): 338–357.
  • Andersen, T. G., Bollerslev, T., Diebold, F. X., and Ebens, H. (2001). “The Distribution of Realized Stock Return Volatility.” Journal of Financial Econometrics, 61: 43–76.
  • Asai, M. and McAleer, M. (2009). “The Structure of Dynamic Correlations in Multivariate Stochastic Volatility Models.” Journal of Econometrics, 150: 182–192.
  • Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A., and Shephard, N. (2009). “Realized Kernels in Practice: Trades and Quotes.” Econometrics Journal, 12(3): C1–C32.
  • — (2011). “Multivariate Realized Kernels: Consistent Positive Semi-Definite Estimators of the Covariation of Equity Prices with Noise and Non-Synchronous Trading.” Journal of Econometrics, 162: 149–169.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2002). “Econometric Analysis of Realized Volatility and Its Use in Estimating Stochastic Volatility.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2): 253–280.
  • Bauer, G. H. and Vorkink, K. (2011). “Forecasting Multivariate Realized Stock Market Volatility.” Journal of Econometrics, 160: 93–101.
  • Bauwens, L., Laurent, S., and Rombouts, J. V. K. (2006). “Multivariate GARCH Models: a Survey.” Journal of Applied Econometrics, 21: 7109.
  • Bollerslev, T. (1986). “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics, 31: 307–327.
  • Brown, R. G. (1959). Statistical Forecasting for Inventory Control. McGraw-Hill.
  • Carter, C. K. and Kohn, R. (1994). “On Gibbs Sampling for State Space Models.” Biometrika, 81: 541–533.
  • Carvalho, C. M., Lopes, H. F., and Aguilar, O. (2011). “Dynamic Stock Selection Strategies: A Structured Factor Model Framework.” In Bayesian Statistics 9. Oxford University Press.
  • Chib, S., Nardari, F., and Shephard, N. (2002). “Markov Chain Monte Carlo Methods for Stochastic Volatility Models.” Journal of Econometrics, 108: 281–316.
  • Chiriac, R. and Voev, V. (2010). “Modelling and Forecasting Multivariate Realized Volatility.” Journal of Applied Econometrics, 26: 922–947.
  • Chiu, T. Y. M., Leonard, T., and Tsui, K.-W. (1996). “The Matrix-Logarithmic Covariance Model.” Journal of the American Statistical Association, 91: 198–210.
  • Choi, C. and Christensen, H. I. (2011). “Robust 3D Visual Tracking Using Particle Filters on the SE(3) Group.” In IEEE International Conference on Robotics and Automation, 4384–4390.
  • Cochrane, J. H. (2005). Asset Pricing. Princeton University Press.
  • Díaz-García, J. A. and Jáimez, R. G. (1997). “Proof of the Conjectures of H. Uhlig on the Singular Multivariate Beta and the Jacobian of a Certain Matrix Transformation.” The Annals of Statistics, 25: 2018–2023.
  • Edelman, A. (2005). “The Mathematics and Applications of (Finite) Random Matrices.” See handouts 1-4. URL http://web.mit.edu/18.325/www/handouts.html
  • Fama, E. F. and French, K. R. (1993). “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics, 33: 3–56.
  • Fox, E. B. and West, M. (2011). “Autoregressive Models for Variance Matrices: Stationary Inverse Wishart Processes.” Technical report, Duke University.
  • Früwirth-Schnatter, S. (1994). “Data Augmentation and Dynamic Linear Models.” Journal of Time Series Analysis, 15: 183–202.
  • Gourieroux, C., Jasiak, J., and Sufana, R. (2009). “The Wishart Autoregressive Process of Multivariate Stochastic Volatility.” Journal of Econometrics, 150: 167–181.
  • Harvey, A., Ruiz, E., and Shephard, N. (1994). “Multivariate Stochastic Volatility Models.” The Review of Economic Studies, 61: 247–264.
  • Hauberg, S., Lauze, F., and Pedersen, K. S. (2013). “Unscented Kalman Filtering on Riemannian Manifolds.” Journal of Mathematical Imaging and Vision, 46: 103–120.
  • Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems For Stochastic Processes. Springer.
  • Jacquier, E., Polson, N. G., and Rossi, P. E. (2004). “Bayesian Analysis of Stochastic Volatility Models with Fat-Tails and Correlated Errors.” Journal of Econometrics, 122: 185–212.
  • Jin, X. and Maheu, J. M. (2012). “Modelling Realized Covariances and Returns.” URL http://homes.chass.utoronto.ca/ jmaheu/jin-maheu.pdf
  • Julier, S. J. and Uhlmann, J. K. (1997). “New Extensions of the Kalman Filter to Nonlinear Systems.” In Signal Processing, Sensor Fusion, and Target Recognition VI, volume 3068.
  • Kalman, R. E. (1960). “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering, 82 (Series D): 35–45.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer.
  • Koopman, S. J., Jungbacker, B., and Hol, E. (2005). “Forecasting Daily Variability of the S&P 100 Stock Index using Historical, Realised and Implied Volatility Measurements.” Journal of Empirical Finance, 12: 445–475.
  • Lence, S. H. and Hayes, D. J. (1994a). “The Empirical Minimum-Variance Hedge.” American Journal of Agricultural Economics, 76: 94–104.
  • — (1994b). “Parameter-Based Decision Making Under Estimation Risk: An Application to Futures Trading.” The Journal of Finance, 49: 345–357.
  • Liu, Q. (2009). “On Portfolio Optimization: How and When Do We Benefit from High-Frequency Data?” Journal of Applied Econometrics, 24: 560–582.
  • Loddo, A., Ni, S., and Sun, D. (2011). “Selection of Multivariate Stochastic Volatility Models via Bayesian Stochastic Search.” Journal of Business and Economic Statistics, 29: 342–355.
  • Mikusiński, P. and Taylor, M. D. (2002). An Introduction to Multivariate Analysis. Birkhäuser.
  • Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley.
  • Philipov, A. and Glickman, M. E. (2006). “Multivariate Stochastic Volatility via Wishart Processes.” Journal of Business and Economic Statistics, 24: 313–328.
  • Pitt, M. K. and Walker, S. G. (2005). “Constructing Stationary Time Series Models Using Auxiliary Variables with Applications.” Journal of the American Statistical Association, 100(470): 554–564.
  • Prado, R. and West, M. (2010). Time Series: Modeling, Computation, and Inference, chapter Multivariate DLMs and Covariance Models, 263–319. Chapman & Hall/CRC.
  • Quintana, J. M. and West, M. (1987). “An Analysis of International Exchange Rates Using Multivariate DLMs.” The Statistician, 36: 275–281.
  • Shephard, N. (1994). “Local Scale Models: State Space Alternative to Integrated GARCH Processes.” Journal of Econometrics, 60: 181–202.
  • — (2005). Stochastic Volatility: Selected Readings. Oxford University Press.
  • Srivastava, A. and Klassen, E. (2004). “Bayesian and Geometric Subspace Tracking.” Advances in Applied Probability, 36(1): 43–56.
  • Taylor, S. J. (1982). Financial Returns Modelled by the Product of Two Stochastic Processes–a Study of Daily Sugar Prices 1961-1979, 203–226. Amersterdam: North-Holland.
  • Tompkins, F. and Wolfe, P. J. (2007). “Bayesian Filtering on the Stiefel Manifold.” In Computational Advances in Multi-Sensor Adaptive Processing, 261 – 264.
  • Triantafyllopoulos, K. (2008). “Multivariate stochastic volatility with Bayesian dynamic linear models.” Journal of Statistical Planning and Inference, 138: 1021–1037.
  • Tyagi, A. and Davis, J. W. (2008). “A Recursive Filter For Linear Systems on Riemannian Manifolds.” In IEEE Conference on Computer Vision and Pattern Recognition.
  • Uhlig, H. (1994). “On Singular Wishart and Singular Multivariate Beta Distributions.” The Annals of Statistics, 22(1): 395–495.
  • — (1997). “Bayesian Vector Autoregressions with Stochastic Volatility.” Econometrica, 65(1): 59–73.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models. Springer Verlag.
  • Windle, J. (2013). “Forecasting High-Dimensional Variance-Covariance Matrices with High-Frequency Data and Sampling Pólya-Gamma Random Variates for Posterior Distributions Derived from Logistic Likelihoods.” Ph.D. thesis, The University of Texas at Austin.
  • Windle, J., Carvalho, C. M., Scott, J. G., and Sun, L. (2013). “Efficient Data Augmentation in Dynamic Models for Binary and Count Data.” URL http://arxiv.org/abs/1308.0774

See also

  • Related item: Roberto Casarin, Comment on Article by Windle and Carvalho. Bayesian Anal., Vol. 9, Iss. 4 (2014) 793–804.
  • 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.