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Bayesian Adaptively Updated Hamiltonian Monte Carlo with an Application to High-Dimensional BEKK GARCH Models

Author

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  • Martin Burda

    () (Department of Economics, University of Toronto, Canada; IES, Charles University, Czech Republic)

  • John M. Maheu

    () (Department of Economics, University of Toronto, Canada; RCEA, Italy)

Abstract

Hamiltonian Monte Carlo (HMC) is a recent statistical procedure to sample from complex distributions. Distant proposal draws are taken in a sequence of steps following the Hamiltonian dynamics of the underlying parameter space, often yielding superior mixing properties of the resulting Markov chain. However, its performance can deteriorate sharply with the degree of irregularity of the underlying likelihood due to its lack of local adaptability in the parameter space. Riemann Manifold HMC (RMHMC), a locally adaptive version of HMC, alleviates this problem, but at a substantially increased computational cost that can become prohibitive in high-dimensional scenarios. In this paper we propose the Adaptively Updated HMC (AUHMC), an alternative inferential method based on HMC that is both fast and locally adaptive, combining the advantages of both HMC and RMHMC. The benefits become more pronounced with higher dimensionality of the parameter space and with the degree of irregularity of the underlying likelihood surface. We show that AUHMC satisfies detailed balance for a valid MCMC scheme and provide a comparison with RMHMC in terms of effective sample size, highlighting substantial efficiency gains of AUHMC. Simulation examples and an application of the BEKK GARCH model show the practical usefulness of the new posterior sampler.

Suggested Citation

  • Martin Burda & John M. Maheu, 2012. "Bayesian Adaptively Updated Hamiltonian Monte Carlo with an Application to High-Dimensional BEKK GARCH Models," Working Paper series 46_12, Rimini Centre for Economic Analysis.
  • Handle: RePEc:rim:rimwps:46_12
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    References listed on IDEAS

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    1. Roman Liesenfeld & Jean-Francois Richard, 2006. "Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 335-360.
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    4. Bauwens, Luc & Bos, Charles S. & van Dijk, Herman K. & van Oest, Rutger D., 2004. "Adaptive radial-based direction sampling: some flexible and robust Monte Carlo integration methods," Journal of Econometrics, Elsevier, vol. 123(2), pages 201-225, December.
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    Cited by:

    1. Audrone Virbickaite & M. Concepción Ausín & Pedro Galeano, 2015. "Bayesian Inference Methods For Univariate And Multivariate Garch Models: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 29(1), pages 76-96, February.
    2. Burda Martin, 2015. "Constrained Hamiltonian Monte Carlo in BEKK GARCH with Targeting," Journal of Time Series Econometrics, De Gruyter, vol. 7(1), pages 1-19, January.

    More about this item

    Keywords

    High-dimensional joint sampling; Markov chain Monte Carlo;

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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