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

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  • 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
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    Citations

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    Cited by:

    1. Jianqing Fan, 2004. "A selective overview of nonparametric methods in financial econometrics," Papers math/0411034, arXiv.org.
    2. Dennis Kristensen, 2007. "Nonparametric Estimation and Misspecification Testing of Diffusion Models," CREATES Research Papers 2007-01, Department of Economics and Business Economics, Aarhus University.
    3. 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.
    4. 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.
    5. Jianqing Fan & Yingying Fan & Jinchi Lv, 0. "Aggregation of Nonparametric Estimators for Volatility Matrix," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 5(3), pages 321-357.
    6. Timothy Christensen, 2014. "Nonparametric Stochastic Discount Factor Decomposition," Papers 1412.4428, arXiv.org, revised May 2017.
    7. Timothy M. Christensen, 2015. "Nonparametric stochastic discount factor decomposition," CeMMAP working papers CWP24/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. 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.
    9. 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.
    10. 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.
    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.

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