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Analytic Convergence Rates and Parameterization Issues for the Gibbs Sampler Applied to State Space Models

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  • Michael K. Pitt
  • Neil Shephard

Abstract

In this paper we obtain a closed form expression for the convergence rate of the Gibbs sampler applied to the unobserved states of a first‐order autoregression plus noise model. The rate is expressed in terms of the parameters of the model, which are regarded as fixed. For the case where the unconditional mean of the states is a parameter of interest we provide evidence that a ‘centred’ parameterization of a state space model is preferable for the performance of the Gibbs sampler. These two results provide guidance when the Gaussianity or linearity of the state space form is lost. We illustrate this by examining the performance of a Markov chain Monte Carlo sampler for the stochastic volatility model.

Suggested Citation

  • Michael K. Pitt & Neil Shephard, 1999. "Analytic Convergence Rates and Parameterization Issues for the Gibbs Sampler Applied to State Space Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 20(1), pages 63-85, January.
    • Handle: RePEc:bla:jtsera:v:20:y:1999:i:1:p:63-85
    DOI: 10.1111/1467-9892.00126
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    1. Kastner, Gregor & Frühwirth-Schnatter, Sylvia, 2014. "Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 408-423.
    2. Steinsland, Ingelin, 2007. "Parallel exact sampling and evaluation of Gaussian Markov random fields," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 2969-2981, March.
    3. Strickland, Chris M. & Martin, Gael M. & Forbes, Catherine S., 2008. "Parameterisation and efficient MCMC estimation of non-Gaussian state space models," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2911-2930, February.
    4. Chan Joshua & Doucet Arnaud & León-González Roberto & Strachan Rodney W., 2025. "Multivariate Stochastic Volatility with Co-Heteroscedasticity," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 29(3), pages 265-300.
    5. Hsun-Jung Cho & Yow-Jen Jou & Chien-Lun Lan, 2009. "Time Dependent Origin-destination Estimation from Traffic Count without Prior Information," Networks and Spatial Economics, Springer, vol. 9(2), pages 145-170, June.
    6. Frühwirth-Schnatter, Sylvia & Wagner, Helga, 2010. "Stochastic model specification search for Gaussian and partial non-Gaussian state space models," Journal of Econometrics, Elsevier, vol. 154(1), pages 85-100, January.
    7. Michael B. Gordy & Pawel J. Szerszen, 2015. "Bayesian Estimation of Time-Changed Default Intensity Models," Finance and Economics Discussion Series 2015-2, Board of Governors of the Federal Reserve System (U.S.).
    8. Vidal-Llana, Xenxo & Uribe, Jorge M. & Guillén, Montserrat, 2023. "European stock market volatility connectedness: The role of country and sector membership," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 82(C).
    9. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 361-393.
    10. Gabriele Fiorentini & Enrique Sentana & Neil Shephard, 2004. "Likelihood-Based Estimation of Latent Generalized ARCH Structures," Econometrica, Econometric Society, vol. 72(5), pages 1481-1517, September.
    11. Strickland, Chris M. & Turner, Ian. W. & Denham, Robert & Mengersen, Kerrie L., 2009. "Efficient Bayesian estimation of multivariate state space models," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4116-4125, October.
    12. Charles Bos & Neil Shephard, 2006. "Inference for Adaptive Time Series Models: Stochastic Volatility and Conditionally Gaussian State Space Form," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 219-244.
    13. Marcin Mider & Paul A. Jenkins & Murray Pollock & Gareth O. Roberts, 2022. "The Computational Cost of Blocking for Sampling Discretely Observed Diffusions," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3007-3027, December.

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