Advertisement

Asymptotic Normality of Convoluted Smoothed Kernel Estimation for Scalar Diffusion Model

  • Yuping SongEmail author
  • Weijie Hou
  • Guang Yang
Article
  • 5 Downloads

Abstract

In this paper, we consider a convoluted smoothed nonparametric approach for the unknown coefficients of diffusion model based on high frequency data. Under regular conditions, we obtain the asymptotic normality for the proposed estimators as the time span T and sample interval Δn → 0. The procedure and asymptotic behavior can be applied for both Harris recurrent and positive Harris recurrent processes. The finite-sample benefits of the underlying estimators are verified through Monte Carlo simulation and 15-min high-frequency stock index in Shanghai Stock Exchange for an empirical application.

Keywords

Diffusion models Volatility function Asymptotic normality Bias and variance reduction Nonstationary high frequency financial data 

Mathematics Subject Classification (2010)

Primary 62G20 62M05 Secondary 60J75 62P20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

This research work is supported by Ministry of Education, Humanities and Social Sciences Project (No. 18YJCZH153), National Statistical Science Research Project (No. 2018LZ05), the General Research Fund of Shanghai Normal University (No. SK201720) and Funding Programs for Youth Teachers of Shanghai Colleges and Universities (No. A-9103-18-104001). The authors would like to thank the editor and two anonymous referees for their valuable suggestions, which greatly improved our paper.

References

  1. Aït-Sahalia Y (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64:527–560CrossRefzbMATHGoogle Scholar
  2. Aït-Sahalia Y, Jacod J (2014) High-frequency financial econometrics. Princeton University Press, PrincetonCrossRefzbMATHGoogle Scholar
  3. Aït-Sahalia Y, Park J (2016) Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models. J Econ 192:119–138MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bandi F, Moloche G (2018) On the functional estimation of multivariate diffusion processes. Economet Theor 34:896–946MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bandi F, Phillips P (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71:241–283MathSciNetCrossRefzbMATHGoogle Scholar
  6. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fan J, Zhang C (2003) A reexamination of diffusion estimators with applications to financial model validation. J Am Stat Assoc 98:118–134MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fan J, Fan Y, Jiang J (2007) Dynamic integration of time- and state-domain methods for volatility estimation. J Am Stat Assoc 102:618–631MathSciNetCrossRefzbMATHGoogle Scholar
  9. Florens-Zmirou D (1993) On estimating the diffusion coefficient from discrete observations. J Appl Probab 30:790–804MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gasser T, Müller H (1979) Kernel estimation of regression function. In: Gasser T, Rosenblatt M (eds) Smoothing techniques for curve estimation. Springer, Heidelberg, pp 23–68Google Scholar
  11. Gospodinovy N, Hirukawa M (2012) Nonparametric estimation of scalar diffusion processes of interest rates using asymmetric kernels. J Empir Financ 19:595–609CrossRefGoogle Scholar
  12. Jacod J (1997) Nonparametric kernel estimation of the diffusion coefficient of a diffusion. Scand J Stat 27:83–96MathSciNetCrossRefzbMATHGoogle Scholar
  13. Jiang G, Knight J (1997) A nonparametric approach to the estimation of diffusion processes, with an application to a short-term interest rate model. Economet Theor 13:615–645MathSciNetCrossRefGoogle Scholar
  14. Karatzas I, Shreve S (2003) Brownian motion and stochastic calculus, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  15. Moloche G (2001) Local nonparametric estimation of scalar diffusions. Working paper, MITGoogle Scholar
  16. Protter P (2004) Stochastic integration and differential equations, 2nd edn. Springer, BerlinGoogle Scholar
  17. Revuz D, Yor M (2005) Continuous martingales and Brownian motion. Springer, New YorkzbMATHGoogle Scholar
  18. Xu Z (2003) Staistical inference for diffusion processes. Ph.D thesis, East China Normal UniversityGoogle Scholar
  19. Xu K (2009) Empirical likelihood based inference for nonparametric recurrent diffusions. J Econ 153:65–82MathSciNetCrossRefzbMATHGoogle Scholar
  20. Xu K (2010) Re-weighted functional estimation of diffusion models. Economet Theor 26:541–563CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Finance and BusinessShanghai Normal UniversityShanghaiPeople’s Republic of China
  2. 2.College of Mathematics and SciencesShanghai Normal UniversityShanghaiPeople’s Republic of China

Personalised recommendations