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Bayesian Analysis of Switching ARCH Models

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  • Sylvia Fruhwirth-Schnattaer

    (University of Economics and Business Administration)

  • Sylvia Kaufmann

    (University of Vienna)

Abstract

We consider a time series model with autoregressive conditional heteroskedasticity that is subject to changes in regime. The regimes evolve according to a multistate latent Markov switching process with unknown transition probabilities, and it is the constant in the variance process of the innovations that is subject to regime shifts. The joint estimation of the latent process and all model parameters is performed within a Bayesian framework using the method of Markov Chain Monte Carlo simulation. One iteration of the sampler involves first a multi-move step to simulate the latent process out of its conditional distribution. The Gibbs sampler can then be used to simulate the parameters, in particular the transition probabilities, for which the full conditional posterior distribution is known. For most parameters, however, the full conditionals do not belong to any well-known family of distributions. The simulations are then based on the Metropolis-Hastings algorithm with carefully chosen proposal densities. We perform model selection with respect to the number of states and the number of autoregressive parameters in the variance process using Bayes factors and model likelihoods. To this aim, the model likelihood is estimated by combining the candidate's formula with importance sampling. The usefulness of the sampler is demonstrated by applying it to the dataset previously used by Hamilton and Susmel who investigated models with switching autoregressive conditional heteroskedasticity using maximum likelihood methods. The paper concludes with some issues related to maximum likelihood methods, to classical model selection, and to potential straightforward extensions of the model presented here.

Suggested Citation

  • Sylvia Fruhwirth-Schnattaer & Sylvia Kaufmann, 2000. "Bayesian Analysis of Switching ARCH Models," Econometric Society World Congress 2000 Contributed Papers 1381, Econometric Society.
  • Handle: RePEc:ecm:wc2000:1381
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    References listed on IDEAS

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    1. Luc Bauwens & Michel Lubrano, 1998. "Bayesian inference on GARCH models using the Gibbs sampler," Econometrics Journal, Royal Economic Society, vol. 1(Conferenc), pages 23-46.
    2. John F. Geweke, 1991. "Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments," Staff Report 148, Federal Reserve Bank of Minneapolis.
    3. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    4. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," Review of Economic Studies, Oxford University Press, vol. 65(3), pages 361-393.
    5. Gray, Stephen F., 1996. "Modeling the conditional distribution of interest rates as a regime-switching process," Journal of Financial Economics, Elsevier, vol. 42(1), pages 27-62, September.
    6. Cai, Jun, 1994. "A Markov Model of Switching-Regime ARCH," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(3), pages 309-316, July.
    7. Chib, Siddhartha, 1996. "Calculating posterior distributions and modal estimates in Markov mixture models," Journal of Econometrics, Elsevier, vol. 75(1), pages 79-97, November.
    8. Hamilton, James D. & Susmel, Raul, 1994. "Autoregressive conditional heteroskedasticity and changes in regime," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 307-333.
    9. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    10. Sylvia Frühwirth-Schnatter, 2001. "Fully Bayesian Analysis of Switching Gaussian State Space Models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 31-49, March.
    11. Nakatsuma, Teruo, 2000. "Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach," Journal of Econometrics, Elsevier, vol. 95(1), pages 57-69, March.
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    Cited by:

    1. Jan Henneke & Svetlozar Rachev & Frank Fabozzi & Metodi Nikolov, 2011. "MCMC-based estimation of Markov Switching ARMA-GARCH models," Applied Economics, Taylor & Francis Journals, vol. 43(3), pages 259-271.
    2. Deschamps, Philippe J., 2006. "A flexible prior distribution for Markov switching autoregressions with Student-t errors," Journal of Econometrics, Elsevier, vol. 133(1), pages 153-190, July.
    3. Lanne, Markku & Luoto, Jani, 2008. "Robustness of the risk-return relationship in the U.S. stock market," Finance Research Letters, Elsevier, vol. 5(2), pages 118-127, June.
    4. Miazhynskaia, Tatiana & Fruhwirth-Schnatter, Sylvia & Dorffner, Georg, 2006. "Bayesian testing for non-linearity in volatility modeling," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 2029-2042, December.
    5. Hyun Kook Shin & Byoung Hark Yoo, 2012. "The Volatility Of The Won-Dollar Exchange Rate During The 2008-9 Crisis," Journal of Economic Development, Chung-Ang Unviersity, Department of Economics, vol. 37(4), pages 61-77, December.
    6. Ardia, David & Hoogerheide, Lennart F., 2010. "Efficient Bayesian estimation and combination of GARCH-type models," MPRA Paper 22919, University Library of Munich, Germany.
    7. Tatiana Miazhynskaia & Georg Dorffner, 2006. "A comparison of Bayesian model selection based on MCMC with an application to GARCH-type models," Statistical Papers, Springer, vol. 47(4), pages 525-549, October.
    8. Mihaela Craioveanu & Eric Hillebrand, 2012. "Level changes in volatility models," Annals of Finance, Springer, vol. 8(2), pages 277-308, May.

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