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Local Nonparametric Estimation of Scalar Diffusions

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  • Moloche, Guillermo

Abstract

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

    as
    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. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    3. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(4), pages 896-946, August.
    4. 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.
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    Citations

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    Cited by:

    1. Xu, Ke-Li, 2010. "Reweighted Functional Estimation Of Diffusion Models," Econometric Theory, Cambridge University Press, vol. 26(2), 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. Bandi, Federico & Corradi, Valentina & Moloche, Guillermo, 2009. "Bandwidth selection for continuous-time Markov processes," MPRA Paper 43682, University Library of Munich, Germany.
    4. 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.
    5. Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series gd08-035, Institute of Economic Research, Hitotsubashi University.
    6. 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.
    7. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(4), pages 896-946, August.
    8. Lu, Zudi & Linton, Oliver, 2007. "Local Linear Fitting Under Near Epoch Dependence," Econometric Theory, Cambridge University Press, vol. 23(1), pages 37-70, February.
    9. Yuping Song & Weijie Hou & Guang Yang, 2020. "Asymptotic Normality of Convoluted Smoothed Kernel Estimation for Scalar Diffusion Model," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 191-221, March.
    10. Wooyong Lee & Priscilla E. Greenwood & Nancy Heckman & Wolfgang Wefelmeyer, 2017. "Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise," Statistical Inference for Stochastic Processes, Springer, vol. 20(2), pages 237-252, July.

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    More about this item

    Keywords

    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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