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Stochastic Volatility: Univariate and Multivariate Extensions

Author

Listed:
  • Eric Jacquier
  • Nicholas G. Polson
  • Peter Rossi

Abstract

Discrete time stochastic volatility models (hereafter SVOL) are noticeably harder to estimate than the successful ARCH family of models. Recent advances in the literature now make it possible to produce efficient estimation and prediction for a basic univariate SVOL model. However, the basic model may be insufficient for numerous economics and finance applications. In this paper, we develop methods for finite sample inference, smoothing, and prediction for a number of univariate and multivariate SVOL models. Specifically, we model fat-tailed and skewed conditional distributions, correlated errors distributions (leverage effect), and two multivariate models, a stochastic factor structure model and a stochastic discount dynamic model. We apply some of these extensions to financial series. We find (1) strong evidence of non-normal conditional distributions for stock returns and exchange rates, and (2) evidence of correlated errors for stock returns. These departures from the basic model affect the measured persistence and hence the prediction of volatility. This result as a policy implication on decisions and models such as asset allocation and option pricing which inputs are prediction of volatility. We specify the models as a hierarchy of conditional distributions: p(data | volatilities), p(volatilities | parameters) and p(parameters). Given a model and the data, inference and prediction are based on the joint posterior distribution of the volatilities and the parameters which we simulate via Markov chain Monte Carlo (hereafter MCMC) methods. This approach also provides a sensitivity analysis for parameter inference and an outlier diagnostic. The hierarchical framework is a natural environment for the construction of SVOL models departing from the standard distributional assumptions.

Suggested Citation

  • Eric Jacquier & Nicholas G. Polson & Peter Rossi, "undated". "Stochastic Volatility: Univariate and Multivariate Extensions," Rodney L. White Center for Financial Research Working Papers 19-95, Wharton School Rodney L. White Center for Financial Research.
  • Handle: RePEc:fth:pennfi:19-95
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    References listed on IDEAS

    as
    1. Geweke, John, 1994. "Priors for Macroeconomic Time Series and Their Application," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 609-632, August.
    2. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
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    5. Eric Ghysels & Andrew Harvey & Éric Renault, 1995. "Stochastic Volatility," CIRANO Working Papers 95s-49, CIRANO.
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    7. 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.
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    9. Friedman, Moshe & Harris, Lawrence, 1998. "A Maximum Likelihood Approach for Non-Gaussian Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 284-291, July.
    10. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
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    12. Ronald Mahieu & Peter Schotman, 1994. "Stochastic volatility and the distribution of exchange rate news," Discussion Paper / Institute for Empirical Macroeconomics 96, Federal Reserve Bank of Minneapolis.
    13. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
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    Citations

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    Cited by:

    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.
    2. Fernández, C. & Steel, M.F.J., 1997. "On the Dangers of Modelling through Continuous Distributions : A Bayesian Perspective," Discussion Paper 1997-05, Tilburg University, Center for Economic Research.
    3. Liesenfeld, Roman & Richard, Jean-Francois, 2003. "Univariate and multivariate stochastic volatility models: estimation and diagnostics," Journal of Empirical Finance, Elsevier, vol. 10(4), pages 505-531, September.
    4. Avouyi-Dovi, S. & Horny, G. & Sevestre, P., 2013. "The dynamics of bank loans short-term interest rates in the Euro area: what lessons can we draw from the current crisis?," Working papers 462, Banque de France.
    5. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility," Microeconomics Working Papers 22058, East Asian Bureau of Economic Research.
    6. Tsyplakov, Alexander, 2010. "Revealing the arcane: an introduction to the art of stochastic volatility models," MPRA Paper 25511, University Library of Munich, Germany.
    7. Andrea Carriero & Todd E. Clark & Massimiliano Marcellino, 2016. "Common Drifting Volatility in Large Bayesian VARs," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 375-390, July.
    8. Meddahi, N., 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    9. Jun Yu & Renate Meyer, 2006. "Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 361-384.
    10. Cogley, Timothy & Morozov, Sergei & Sargent, Thomas J., 2005. "Bayesian fan charts for U.K. inflation: Forecasting and sources of uncertainty in an evolving monetary system," Journal of Economic Dynamics and Control, Elsevier, vol. 29(11), pages 1893-1925, November.
    11. Norberto Rodríguez, 2000. "Bayesian Model Estimation and Selection for the Weekly Colombian Exchange Rate," Borradores de Economia 161, Banco de la Republica de Colombia.
    12. Ishihara, Tsunehiro & Omori, Yasuhiro, 2012. "Efficient Bayesian estimation of a multivariate stochastic volatility model with cross leverage and heavy-tailed errors," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3674-3689.
    13. Danielsson, Jon, 1998. "Multivariate stochastic volatility models: Estimation and a comparison with VGARCH models," Journal of Empirical Finance, Elsevier, vol. 5(2), pages 155-173, June.
    14. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
    15. Fern ndez, Carmen & Steel, Mark F.J., 2000. "Bayesian Regression Analysis With Scale Mixtures Of Normals," Econometric Theory, Cambridge University Press, vol. 16(01), pages 80-101, February.
    16. Stavros Degiannakis & Alexandra Livada & Epaminondas Panas, 2008. "Rolling-sampled parameters of ARCH and Levy-stable models," Applied Economics, Taylor & Francis Journals, vol. 40(23), pages 3051-3067.
    17. repec:pit:wpaper:322 is not listed on IDEAS
    18. Timothy Cogley & Thomas J. Sargent, 2005. "Drift and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 8(2), pages 262-302, April.
    19. repec:bla:irvfin:v:17:y:2017:i:3:p:479-490 is not listed on IDEAS
    20. Mikhail Chernov & Eric Ghysels, 1998. "What Data Should Be Used to Price Options?," CIRANO Working Papers 98s-22, CIRANO.
    21. Timothy Cogley, 2005. "Changing Beliefs and the Term Structure of Interest Rates: Cross-Equation Restrictions with Drifting Parameters," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 8(2), pages 420-451, April.
    22. Sassan Alizadeh & Michael W. Brandt & Francis X. Diebold, 2001. "High- and Low-Frequency Exchange Rate Volatility Dynamics: Range-Based Estimation of Stochastic Volatility Models," NBER Working Papers 8162, National Bureau of Economic Research, Inc.

    More about this item

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • 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
    • G1 - Financial Economics - - General Financial Markets

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