IDEAS home Printed from https://ideas.repec.org/p/zbw/sfb373/200257.html
   My bibliography  Save this paper

Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed

Author

Listed:
  • Gobet, Emmanuel
  • Hoffmann, Marc
  • Reiß, Markus

Abstract

We study the problem of estimating the coefficients of a diffusion (Xl, t 2: 0); the estimation is based on discrete data Xn . . n = 0, 1, ... ,N. The sampling frequency delta t is constant , and asymptotics arc taken at the number of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient - the volatility - and the drift in a nonparametric setting is ill-posed: The minimax rates of convergence for Sobolev constraints and squared-crror lOBS coincide with that of a respectively first and second order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. An important consequence of this result is that we can characterize quantitatively the difference between the estimation of a diffusion in the low frequency sampling case and inference problems in other related frameworks: nonparametric estimation of a diffusion based on continuous or high frequency data: but also parametric estimation for fixed delta, in which case root-N-consistent estimators usually exist. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X/lA, n = 0,1, ... ,N) in a suitable Sobolev norm: together with an estimation of its invariant density.

Suggested Citation

  • Gobet, Emmanuel & Hoffmann, Marc & Reiß, Markus, 2002. "Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed," SFB 373 Discussion Papers 2002,57, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  • Handle: RePEc:zbw:sfb373:200257
    as

    Download full text from publisher

    File URL: https://www.econstor.eu/bitstream/10419/65321/1/72702194X.pdf
    Download Restriction: no
    ---><---

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dennis Kristensen, 2007. "Nonparametric Estimation and Misspecification Testing of Diffusion Models," CREATES Research Papers 2007-01, Department of Economics and Business Economics, Aarhus University.
    2. Timothy Christensen, 2014. "Nonparametric Stochastic Discount Factor Decomposition," Papers 1412.4428, arXiv.org, revised May 2017.
    3. Kristensen, Dennis, 2004. "Estimation in two classes of semiparametric diffusion models," LSE Research Online Documents on Economics 24739, London School of Economics and Political Science, LSE Library.
    4. Jianqing Fan, 2004. "A selective overview of nonparametric methods in financial econometrics," Papers math/0411034, arXiv.org.
    5. Jianqing Fan & Yingying Fan & Jinchi Lv, 0. "Aggregation of Nonparametric Estimators for Volatility Matrix," Journal of Financial Econometrics, Oxford University Press, vol. 5(3), pages 321-357.
    6. Timothy M. Christensen, 2015. "Nonparametric stochastic discount factor decomposition," CeMMAP working papers CWP24/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Ilia Negri & Yoichi Nishiyama, 2011. "Goodness of fit test for small diffusions by discrete time observations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(2), pages 211-225, April.
    8. De Gregorio, Alessandro & Maria Iacus, Stefano, 2010. "Clustering of discretely observed diffusion processes," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 598-606, February.
    9. Ilia Negri & Yoichi Nishiyama, 2010. "Goodness of fit test for ergodic diffusions by tick time sample scheme," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 81-95, April.
    10. Xiaohong Chen & Lars Peter Hansen & Jose Scheinkman, 2009. "Principal Components and Long Run Implications of Multivariate Diffusions," Cowles Foundation Discussion Papers 1694, Cowles Foundation for Research in Economics, Yale University.
    11. Yunyan Wang & Lixin Zhang & Mingtian Tang, 2012. "Re-weighted functional estimation of second-order diffusion processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1129-1151, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:sfb373:200257. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ZBW - Leibniz Information Centre for Economics (email available below). General contact details of provider: https://edirc.repec.org/data/sfhubde.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.