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Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models

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  • Dennis Kristensen

    (Department of Economics, University of Columbia, NY)

Abstract

We propose novel misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in Kristensen (Journal of Econometrics, 2010). We first demonstrate that given a preliminary estimator of either the drift or the diffusion term in a diffusion model, nonparametric kernel estimators of the remaining term can be obtained. We then propose misspecification tests of semparametric and fully parametric diffusion models that compare estimators of the transition density under the relevant null and alternative. The asymptotic distribution of the estimators and tests under the null are derived, and the power properties are analyzed by considering contiguous alternatives. Test directly comparing the drift and diffusion estimators under the relevant null and alternative are also analyzed. Markov Bootstrap versions of the test statistics are proposed to improve on the finite-sample approximations. The finite sample properties of the estimators are examined in a simulation study.

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Bibliographic Info

Paper provided by University of Copenhagen. Department of Economics in its series Discussion Papers with number 10-10.

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Length: 17 pages
Date of creation: Mar 2010
Date of revision:
Handle: RePEc:kud:kuiedp:1010

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Related research

Keywords: diffusion process; kernel estimation; nonparametric; specification testing; semiparametric; transition density;

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  1. Federico M. Bandi & Peter C.B. Phillips, 2005. "A Simple Approach to the Parametric Estimation of Potentially Nonstationary Diffusions," Cowles Foundation Discussion Papers 1522, Cowles Foundation for Research in Economics, Yale University.
  2. Gourieroux, C. & Monfort, A. & Renault, E., 1992. "Indirect Inference," Papers 92.279, Toulouse - GREMAQ.
  3. Joel L. Horowitz, 2003. "Bootstrap Methods for Markov Processes," Econometrica, Econometric Society, vol. 71(4), pages 1049-1082, 07.
  4. Kristensen, Dennis & Shin, Yongseok, 2012. "Estimation of dynamic models with nonparametric simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 167(1), pages 76-94.
  5. Dennis Kristensen, 2008. "Uniform Convergence Rates of Kernel Estimators with Heterogenous, Dependent Data," CREATES Research Papers 2008-37, School of Economics and Management, University of Aarhus.
  6. Andrews, Donald W K, 1994. "Asymptotics for Semiparametric Econometric Models via Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 62(1), pages 43-72, January.
  7. Dennis Kristensen, 2009. "Pseudo-Maximum Likelihood Estimation in Two Classes of Semiparametric Diffusion Models," CREATES Research Papers 2009-41, School of Economics and Management, University of Aarhus.
  8. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
  9. Escanciano, J. Carlos, 2009. "On The Lack Of Power Of Omnibus Specification Tests," Econometric Theory, Cambridge University Press, vol. 25(01), pages 162-194, February.
  10. Bhardwaj, Geetesh & Corradi, Valentina & Swanson, Norman R., 2008. "A Simulation-Based Specification Test for Diffusion Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 176-193, April.
  11. Wooldridge, Jeffrey M. & White, Halbert, 1988. "Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes," Econometric Theory, Cambridge University Press, vol. 4(02), pages 210-230, August.
  12. Chen, Xiaohong & Hansen, Lars Peter & Carrasco, Marine, 2008. "Nonlinearity and Temporal Dependence," Working Papers 48, Yale University, Department of Economics.
  13. Dennis Kristensen, 2007. "Nonparametric Estimation and Misspecification Testing of Diffusion Models," CREATES Research Papers 2007-01, School of Economics and Management, University of Aarhus.
  14. Gouriéroux, Christian & Tenreiro, Carlos, 2001. "Local Power Properties of Kernel Based Goodness of Fit Tests," Journal of Multivariate Analysis, Elsevier, vol. 78(2), pages 161-190, August.
  15. Nicolau, Jo o, 2003. "Bias Reduction In Nonparametric Diffusion Coefficient Estimation," Econometric Theory, Cambridge University Press, vol. 19(05), pages 754-777, October.
  16. Chunrong Ai, 1997. "A Semiparametric Maximum Likelihood Estimator," Econometrica, Econometric Society, vol. 65(4), pages 933-964, July.
  17. Fan, Yanqin, 1994. "Testing the Goodness of Fit of a Parametric Density Function by Kernel Method," Econometric Theory, Cambridge University Press, vol. 10(02), pages 316-356, June.
  18. Corradi, Valentina & Swanson, Norman R., 2005. "Bootstrap specification tests for diffusion processes," Journal of Econometrics, Elsevier, vol. 124(1), pages 117-148, January.
  19. Whang, Yoon-Jae & Andrews, Donald W. K., 1993. "Tests of specification for parametric and semiparametric models," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 277-318.
  20. Li, Fuchun & Tkacz, Greg, 2006. "A consistent bootstrap test for conditional density functions with time-series data," Journal of Econometrics, Elsevier, vol. 133(2), pages 863-886, August.
  21. Ilia Negri & Yoichi Nishiyama, 2009. "Goodness of fit test for ergodic diffusion processes," Annals of the Institute of Statistical Mathematics, Springer, vol. 61(4), pages 919-928, December.
  22. 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.
  23. Aït-Sahalia, Yacine & Fan, Jianqing & Peng, Heng, 2009. "Nonparametric Transition-Based Tests for Jump Diffusions," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1102-1116.
  24. Li, Fuchun, 2007. "Testing The Parametric Specification Of The Diffusion Function In A Diffusion Process," Econometric Theory, Cambridge University Press, vol. 23(02), pages 221-250, April.
  25. Chen, Song Xi & Gao, Jiti & Tang, Chenghong, 2005. "A test for model specification of diffusion processes," MPRA Paper 11976, University Library of Munich, Germany, revised Feb 2007.
  26. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(03), pages 560-586, June.
  27. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
  28. Kristensen, Dennis, 2008. "Estimation of partial differential equations with applications in finance," Journal of Econometrics, Elsevier, vol. 144(2), pages 392-408, June.
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