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The Pearson diffusions: A class of statistically tractable diffusion processes

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  • Michael Sørensen
  • Julie Lyng Forman

    () (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating func- tions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-likelihood, and non- linear weighted least squares estimation too, and it is discussed how explicit likeli- hood or approximate likelihood inference is possible for the Pearson diffusions. A complete model classification is presented for the ergodic Pearson diffusions. The class of stationary distributions equals the full Pearson system of distributions. Well-known instances are the Ornstein-Uhlenbeck processes and the square root (CIR) processes. Also diffusions with heavy-tailed and skew marginals are included. Special attention is given to a skew t-type distribution. Explicit formulae for the conditional moments and the polynomial eigenfunctions are derived. The analyti- cal tractability is inherited by transformed Pearson diffusions, integrated Pearson diffusions, sums of Pearson diffusions, and stochastic volatility models with Pearson volatility process. For the non-Markov models explicit optimal prediction based estimating functions are found and shown to yield consistent and asymptotically normal estimators.

Suggested Citation

  • Michael Sørensen & Julie Lyng Forman, 2007. "The Pearson diffusions: A class of statistically tractable diffusion processes," CREATES Research Papers 2007-28, Department of Economics and Business Economics, Aarhus University.
  • Handle: RePEc:aah:create:2007-28
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Nina Munkholt Jakobsen & Michael Sørensen, 2015. "Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval," CREATES Research Papers 2015-33, Department of Economics and Business Economics, Aarhus University.
    2. Aleksandar Mijatovic & Paul Schneider, 2009. "Empirical Asset Pricing with Nonlinear Risk Premia," Working Papers wp09-03, Warwick Business School, Finance Group.
    3. Almut Veraart & Luitgard Veraart, 2012. "Stochastic volatility and stochastic leverage," Annals of Finance, Springer, vol. 8(2), pages 205-233, May.
    4. Hanson, Gordon H. & Lind, Nelson & Muendler, Marc-Andreas, 2015. "The Dynamics of Comparative Advantage," CAGE Online Working Paper Series 252, Competitive Advantage in the Global Economy (CAGE).
    5. Larsson, Martin & Pulido, Sergio, 2017. "Polynomial diffusions on compact quadric sets," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 901-926.
    6. Pierre Blanc & Jonathan Donier & Jean-Philippe Bouchaud, 2015. "Quadratic Hawkes processes for financial prices," Papers 1509.07710, arXiv.org.
    7. Damir Filipovi'c & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Papers 1711.08043, arXiv.org.
    8. Asger Lunde & Anne Floor Brix, 2013. "Estimating Stochastic Volatility Models using Prediction-based Estimating Functions," CREATES Research Papers 2013-23, Department of Economics and Business Economics, Aarhus University.
    9. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    10. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    11. repec:eee:spapps:v:127:y:2017:i:11:p:3512-3535 is not listed on IDEAS
    12. repec:eee:ejores:v:266:y:2018:i:3:p:1153-1174 is not listed on IDEAS
    13. Bercu, Bernard & Richou, Adrien, 2017. "Large deviations for the Ornstein–Uhlenbeck process without tears," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 45-55.
    14. Michael Sørensen, 2008. "Parametric inference for discretely sampled stochastic differential equations," CREATES Research Papers 2008-18, Department of Economics and Business Economics, Aarhus University.
    15. Yuichi Nagahara, 2008. "A Method of Calculating the Downside Risk by Multivariate Nonnormal Distributions," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 15(3), pages 175-184, December.
    16. Giorgos Sermaidis & Omiros Papaspiliopoulos & Gareth O. Roberts & Alexandros Beskos & Paul Fearnhead, 2013. "Markov Chain Monte Carlo for Exact Inference for Diffusions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 294-321, June.
    17. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    18. Yuichi Nagahara, 2011. "Using Nonnormal Distributions to Analyze the Relationship Between Stock Returns in Japan and the US," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(4), pages 429-443, November.

    More about this item

    Keywords

    eigenfunction; ergodic diffusion; integrated diffusion; martingale estimating function; likelihood inference; mixing; optimal estimating function; Pearson system; prediction based estimating function; quasi likelihood; spectral methods; stochastic differential equation; stochastic volatility;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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