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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.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 46154.

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Date of creation: 23 Sep 2001
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Handle: RePEc:pra:mprapa:46154

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Keywords: Nonparametric estimation; Diffusion processes;

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References

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  1. Zirbel, Craig L., 1997. "Mean occupation times of continuous one-dimensional Markov processes," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 69(2), pages 161-178, September.
  2. Bandi, Federico & Moloche, Guillermo, 2008. "On the functional estimation of multivariate diffusion processes," MPRA Paper 43681, University Library of Munich, Germany.
  3. Yakowitz, Sidney, 1989. "Nonparametric density and regression estimation for Markov sequences without mixing assumptions," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 30(1), pages 124-136, July.
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Citations

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Cited by:
  1. Bandi, Federico & Moloche, Guillermo, 2008. "On the functional estimation of multivariate diffusion processes," MPRA Paper 43681, University Library of Munich, Germany.
  2. Bandi, Federico & Corradi, Valentina & Moloche, Guillermo, 2009. "Bandwidth selection for continuous-time Markov processes," MPRA Paper 43682, University Library of Munich, Germany.
  3. Lu, Zudi & Linton, Oliver, 2007. "Local Linear Fitting Under Near Epoch Dependence," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 23(01), pages 37-70, February.
  4. Xu, Ke-Li & Phillips, Peter C. B., 2011. "Tilted Nonparametric Estimation of Volatility Functions With Empirical Applications," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 29(4), pages 518-528.
  5. Peter C.B. Phillips & Ke-Li Xu, 2007. "Tilted Nonparametric Estimation of Volatility Functions," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1612, Cowles Foundation for Research in Economics, Yale University, revised Jul 2010.
  6. Xu, Ke-Li, 2010. "Reweighted Functional Estimation Of Diffusion Models," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 26(02), pages 541-563, April.
  7. Xu, Ke-Li, 2009. "Empirical likelihood-based inference for nonparametric recurrent diffusions," Journal of Econometrics, Elsevier, Elsevier, vol. 153(1), pages 65-82, November.
  8. Federico M. Bandi & Roberto Reno, 2009. "Nonparametric Stochastic Volatility," Global COE Hi-Stat Discussion Paper Series, Institute of Economic Research, Hitotsubashi University gd08-035, Institute of Economic Research, Hitotsubashi University.

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