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Local Nonparametric Estimation of Scalar Diffusions


  • Moloche, Guillermo


This paper studies the functional estimation of the drift and diffusion functions for recurrent scalar diffusion processes from equally spaced observations using the local polynomial kernel approach. Almost sure convergence and a CLT for the estimators are established as the sampling frequency and the time span go to infinity. The asymptotic distributions follow a mixture of normal laws. This theory covers both positive and null recurrent diffusions. Almost sure convergence rates are sometimes path dependent but expected rates can always be characterized in terms of regularly varying functions. The general theory is specialized for positive recurrent diffusion processes, and it is shown in this case that the asymptotic distributions are normal. We also obtain the limit theory for kernel density estimators when the process is positive recurrent, namely, requiring only that the invariant probability measure exists. Nonetheless, it is also shown that such an estimator paradoxically vanishes almost surely when the invariant measure is fat tailed and nonintegrable, that is, in the null recurrent case.

Suggested Citation

  • Moloche, Guillermo, 2001. "Local Nonparametric Estimation of Scalar Diffusions," MPRA Paper 46154, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:46154

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    References listed on IDEAS

    1. Yakowitz, Sidney, 1989. "Nonparametric density and regression estimation for Markov sequences without mixing assumptions," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 124-136, July.
    2. repec:cup:etheor:v:34:y:2018:i:04:p:896-946_00 is not listed on IDEAS
    3. Zirbel, Craig L., 1997. "Mean occupation times of continuous one-dimensional Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 69(2), pages 161-178, September.
    4. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(04), pages 896-946, August.
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    Cited by:

    1. Xu, Ke-Li, 2010. "Reweighted Functional Estimation Of Diffusion Models," Econometric Theory, Cambridge University Press, vol. 26(02), pages 541-563, April.
    2. Xu, Ke-Li, 2009. "Empirical likelihood-based inference for nonparametric recurrent diffusions," Journal of Econometrics, Elsevier, vol. 153(1), pages 65-82, November.
    3. repec:cup:etheor:v:34:y:2018:i:04:p:896-946_00 is not listed on IDEAS
    4. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(04), pages 896-946, August.
    5. Lu, Zudi & Linton, Oliver, 2007. "Local Linear Fitting Under Near Epoch Dependence," Econometric Theory, Cambridge University Press, vol. 23(01), pages 37-70, February.
    6. Bandi, Federico & Corradi, Valentina & Moloche, Guillermo, 2009. "Bandwidth selection for continuous-time Markov processes," MPRA Paper 43682, University Library of Munich, Germany.
    7. Ke-Li Xu & Peter C. B. Phillips, 2011. "Tilted Nonparametric Estimation of Volatility Functions With Empirical Applications," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(4), pages 518-528, October.
    8. repec:spr:sistpr:v:20:y:2017:i:2:d:10.1007_s11203-016-9141-5 is not listed on IDEAS
    9. Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
    10. Peter C.B. Phillips & Ke-Li Xu, 2007. "Tilted Nonparametric Estimation of Volatility Functions," Cowles Foundation Discussion Papers 1612, Cowles Foundation for Research in Economics, Yale University, revised Jul 2010.

    More about this item


    Nonparametric estimation; Diffusion processes;

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics


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