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Likelihood-based inference for a class of multivariate diffusions with unobserved paths

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  • Kalogeropoulos, Konstantinos

Abstract

This paper presents a Markov chain Monte Carlo algorithm for a class of multivariate diffusion models with unobserved paths. This class is of high practical interest as it includes most diffusion driven stochastic volatility models. The algorithm is based on a data augmentation scheme where the paths are treated as missing data. However, unless these paths are transformed so that the dominating measure is independent of any parameters, the algorithm becomes reducible. The methodology developed in Roberts and Stramer (2001 Biometrika 88(3):603-621) circumvents the problem for scalar diffusions. We extend this framework to the class of models of this paper by introducing an appropriate reparametrisation of the likelihood that can be used to construct an irreducible data augmentation scheme. Practical implementation issues are considered and the methodology is applied to simulated data from the Heston model.

Suggested Citation

  • Kalogeropoulos, Konstantinos, 2007. "Likelihood-based inference for a class of multivariate diffusions with unobserved paths," LSE Research Online Documents on Economics 31423, London School of Economics and Political Science, LSE Library.
  • Handle: RePEc:ehl:lserod:31423
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    File URL: http://eprints.lse.ac.uk/31423/
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    References listed on IDEAS

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    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    2. Chernov, Mikhail & Ronald Gallant, A. & Ghysels, Eric & Tauchen, George, 2003. "Alternative models for stock price dynamics," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 225-257.
    3. Elerain, Ola & Chib, Siddhartha & Shephard, Neil, 2001. "Likelihood Inference for Discretely Observed Nonlinear Diffusions," Econometrica, Econometric Society, vol. 69(4), pages 959-993, July.
    4. Christopher S. Jones, 2003. "Nonlinear Mean Reversion in the Short-Term Interest Rate," Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 793-843, July.
    5. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    6. Durham, Garland B & Gallant, A Ronald, 2002. "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 297-316, July.
    7. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    8. Darrell Duffie & Rui Kan, 1996. "A Yield‐Factor Model Of Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 379-406, October.
    9. Shephard, Neil (ed.), 2005. "Stochastic Volatility: Selected Readings," OUP Catalogue, Oxford University Press, number 9780199257201.
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    Citations

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    Cited by:

    1. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    2. Konstantinos Kalogeropoulos & Gareth O. Roberts & Petros Dellaportas, 2007. "Inference for stochastic volatility models using time change transformations," Papers 0711.1594, arXiv.org.
    3. Konstantinos Kalogeropoulos & Petros Dellaportas & Gareth O. Roberts, 2007. "Likelihood-based inference for correlated diffusions," Papers 0711.1595, arXiv.org.
    4. Huang Xiao, 2013. "Quasi-maximum likelihood estimation of multivariate diffusions," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(2), pages 179-197, April.
    5. Beskos, Alexandros & Kalogeropoulos, Konstantinos & Pazos, Erik, 2013. "Advanced MCMC methods for sampling on diffusion pathspace," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1415-1453.
    6. Golightly, A. & Wilkinson, D.J., 2008. "Bayesian inference for nonlinear multivariate diffusion models observed with error," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1674-1693, January.
    7. Dureau, Joseph & Kalogeropoulos, Konstantinos & Baguelin, Marc, 2013. "Capturing the time-varying drivers of an epidemic using stochastic dynamical systems," LSE Research Online Documents on Economics 41749, London School of Economics and Political Science, LSE Library.

    More about this item

    JEL classification:

    • 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
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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