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Monte Carlo Estimation for Nonlinear Non-Gaussian State Space Models

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  • Borus Jungbacker
  • Siem Jan Koopman

Abstract

We develop a proposal or importance density for state space models with a nonlinear non-Gaussian observation vector y ∼ p(y¦θ) and an unobserved linear Gaussian signal vector θ ∼ p(θ). The proposal density is obtained from the Laplace approximation of the smoothing density p(θ¦y). We present efficient algorithms to calculate the mode of p(θ¦y) and to sample from the proposal density. The samples can be used for importance sampling and Markov chain Monte Carlo methods. The new results allow the application of these methods to state space models where the observation density p(y¦θ) is not log-concave. Additional results are presented that lead to computationally efficient implementations. We illustrate the methods for the stochastic volatility model with leverage. Copyright 2007, Oxford University Press.

Suggested Citation

  • Borus Jungbacker & Siem Jan Koopman, 2007. "Monte Carlo Estimation for Nonlinear Non-Gaussian State Space Models," Biometrika, Biometrika Trust, vol. 94(4), pages 827-839.
    • Handle: RePEc:oup:biomet:v:94:y:2007:i:4:p:827-839
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    File URL: http://hdl.handle.net/10.1093/biomet/asm074
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    Cited by:

    1. Siem Jan Koopman & Rutger Lit, 2015. "A dynamic bivariate Poisson model for analysing and forecasting match results in the English Premier League," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 178(1), pages 167-186, January.
    2. Tsyplakov, Alexander, 2010. "The links between inflation and inflation uncertainty at the longer horizon," MPRA Paper 26908, University Library of Munich, Germany.
    3. Christopher Wikle & Mevin Hooten, 2010. "A general science-based framework for dynamical spatio-temporal models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 417-451, November.
    4. Bart Keijsers & Bart Diris & Erik Kole, 2018. "Cyclicality in losses on bank loans," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(4), pages 533-552, June.
    5. Bretó, Carles, 2014. "On idiosyncratic stochasticity of financial leverage effects," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 20-26.
    6. Siem Jan Koopman & Rutger Lit & Thuy Minh Nguyen, 2012. "Fast Efficient Importance Sampling by State Space Methods," Tinbergen Institute Discussion Papers 12-008/4, Tinbergen Institute, revised 16 Oct 2014.
    7. Rutger Jan Lange, 2020. "Bellman filtering for state-space models," Tinbergen Institute Discussion Papers 20-052/III, Tinbergen Institute, revised 19 May 2021.
    8. Lange, Rutger-Jan, 2024. "Bellman filtering and smoothing for state–space models," Journal of Econometrics, Elsevier, vol. 238(2).
    9. Geert Mesters & Victor van der Geest & Catrien Bijleveld, 2014. "Crime, Employment and Social Welfare: an Individual-level Study on Disadvantaged Males," Tinbergen Institute Discussion Papers 14-091/III, Tinbergen Institute.
    10. Siem Jan Koopman & André Lucas & Marcel Scharth, 2016. "Predicting Time-Varying Parameters with Parameter-Driven and Observation-Driven Models," The Review of Economics and Statistics, MIT Press, vol. 98(1), pages 97-110, March.
    11. Tsyplakov Alexander, 2010. "The links between inflation and inflation uncertainty at the longer horizon," EERC Working Paper Series 10/09e, EERC Research Network, Russia and CIS.
    12. Siem Jan Koopman & André Lucas & Marcel Scharth, 2015. "Numerically Accelerated Importance Sampling for Nonlinear Non-Gaussian State-Space Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(1), pages 114-127, January.
    13. Alexander Tsyplakov, 2011. "An introduction to state space modeling (in Russian)," Quantile, Quantile, issue 9, pages 1-24, July.
    14. G. Mesters & S. J. Koopman & M. Ooms, 2016. "Monte Carlo Maximum Likelihood Estimation for Generalized Long-Memory Time Series Models," Econometric Reviews, Taylor & Francis Journals, vol. 35(4), pages 659-687, April.
    15. Charles S. Bos, 2011. "Relating Stochastic Volatility Estimation Methods," Tinbergen Institute Discussion Papers 11-049/4, Tinbergen Institute.
    16. Koopman, Siem Jan & Shephard, Neil & Creal, Drew, 2009. "Testing the assumptions behind importance sampling," Journal of Econometrics, Elsevier, vol. 149(1), pages 2-11, April.
    17. S. J. Koopman & G. Mesters, 2017. "Empirical Bayes Methods for Dynamic Factor Models," The Review of Economics and Statistics, MIT Press, vol. 99(3), pages 486-498, July.
    18. 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.
    19. Mesters, G. & Koopman, S.J., 2014. "Generalized dynamic panel data models with random effects for cross-section and time," Journal of Econometrics, Elsevier, vol. 180(2), pages 127-140.
    20. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    21. Wu, Xin-Yu & Ma, Chao-Qun & Wang, Shou-Yang, 2012. "Warrant pricing under GARCH diffusion model," Economic Modelling, Elsevier, vol. 29(6), pages 2237-2244.
    22. Smith, Michael Stanley & Maneesoonthorn, Worapree, 2018. "Inversion copulas from nonlinear state space models with an application to inflation forecasting," International Journal of Forecasting, Elsevier, vol. 34(3), pages 389-407.
    23. Caterina Schiavoni & Siem Jan Koopman & Franz Palm & Stephan Smeekes & Jan van den Brakel, 2021. "Time-varying state correlations in state space models and their estimation via indirect inference," Tinbergen Institute Discussion Papers 21-020/III, Tinbergen Institute.
    24. Carles Bret'o, 2013. "On idiosyncratic stochasticity of financial leverage effects," Papers 1312.5496, arXiv.org.
    25. Kleppe, Tore Selland & Skaug, Hans Julius, 2012. "Fitting general stochastic volatility models using Laplace accelerated sequential importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3105-3119.

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