Pre-averaged kernel estimators for the drift function of a diffusion process in the presence of microstructure noise

  • Wooyong Lee
  • Priscilla E. Greenwood
  • Nancy Heckman
  • Wolfgang Wefelmeyer


We consider estimation of the drift function of a stationary diffusion process when we observe high-frequency data with microstructure noise over a long time interval. We propose to estimate the drift function at a point by a Nadaraya–Watson estimator that uses observations that have been pre-averaged to reduce the noise. We give conditions under which our estimator is consistent and asympotically normal. Its rate and asymptotic bias and variance are the same as those without microstructure noise. To use our method in data analysis, we propose a data-based cross-validation method to determine the bandwidth in the Nadaraya–Watson estimator. Via simulation, we study several methods of bandwidth choices, and compare our estimator to several existing estimators. In terms of mean squared error, our new estimator outperforms existing estimators.


Diffusion process Nonparametric estimation Discrete observations High-frequency observations Microstructure noise Pre-averaging Drift estimation Nadaraya–Watson estimator 



The research of W. Lee and N. Heckman was supported by the Natural Sciences and Engineering Research Council of Canada. We thank the referees for reading the paper carefully and for suggesting a number of improvements.


  1. Aït-Sahalia Y (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64:527–560CrossRefMATHGoogle Scholar
  2. Arfi M (1995) Nonparametric drift estimation from ergodic samples. J. Nonparametr. Statist. 5:381–389MathSciNetCrossRefMATHGoogle Scholar
  3. Bandi FM, Nguyen TH (2003) On the functional estimation of jump-diffusion models. J. Econometrics 116:293–328MathSciNetCrossRefMATHGoogle Scholar
  4. Bandi FM, Phillips PCB (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71:241–283MathSciNetCrossRefMATHGoogle Scholar
  5. Bandi FM, Phillips PCB (2009a) Nonstationary continuous-time processes. In: Aït-Sahalia Y, Hansen LP (eds) Handbook of Financial Econometrics, vol 1. North-Holland/Elsevier, Amsterdam, pp 139–201Google Scholar
  6. Bandi FM, Corradi V, Moloche G (2009b) Bandwidth selection for continuous time Markov processes. Department of Economics, University of Warwick, PreprintGoogle Scholar
  7. Bhan C, Mandrekar V (2010) Recurrence properties of term structure models. Int. J. Contemp. Math. Sci 5(33–36):1645–1652MathSciNetMATHGoogle Scholar
  8. Burman P, Chow E, Nolan D (1994) A cross-validatory method for dependent data. Biometrika 81:351–358MathSciNetCrossRefMATHGoogle Scholar
  9. Chapman DA, Pearson ND (2000) Is the short rate drift actually nonlinear? J. Finance 55:355–388CrossRefGoogle Scholar
  10. Chu C-K, Marron JS (1991) Comparison of two bandwidth selectors with dependent errors. Ann. Statist. 19:1906–1918MathSciNetCrossRefMATHGoogle Scholar
  11. Comte F, Genon-Catalot V, Rozenholc Y (2007) Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13:514–543MathSciNetCrossRefMATHGoogle Scholar
  12. Comte F, Genon-Catalot V, Rozenholc Y (2012) Non-parametric estimation of the coefficients of ergodic diffusion processes based on high-frequency data. In: Statistical Methods for Stochastic Differential Equations, (M. Kessler, A. Lindner and M. Sørensen, eds.), 341–381, Monogr. Statist. Appl. Probab. 124, CRC Press, Boca RatonGoogle Scholar
  13. Dalalyan A (2005) Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33:2507–2528MathSciNetCrossRefMATHGoogle Scholar
  14. Dalalyan AS, Kutoyants YA (2002) Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Methods Statist. 1:402–427MathSciNetGoogle Scholar
  15. Friz P, Victoir N (2005) Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré Probab. Statist. 41:703–724MathSciNetCrossRefMATHGoogle Scholar
  16. Hoffmann M (1999) Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79:135–163MathSciNetCrossRefMATHGoogle Scholar
  17. Iacus SM (2008) Simulation and Inference for Stochastic Differential Equations. With R examples. Springer Series in Statistics. Springer, New YorkCrossRefMATHGoogle Scholar
  18. Iacus SM (2014) sde: Simulation and Inference for Stochastic Differential Equations. R package version 2.0.13.
  19. Jacod J, Li Y, Mykland PA, Podolskij M, Vetter M (2009) Microstructure noise in the continuous case: the pre-averaging approach. Stochastic Process. Appl. 119:2249–2276MathSciNetCrossRefMATHGoogle Scholar
  20. Jacod J, Protter P (2012) Discretization of Processes. Stochastic Modelling and Applied Probability, vol 67. Springer, HeidelbergMATHGoogle Scholar
  21. Jin P, Mandrekar V, Rüdiger B, Trabelsi C (2013) Positive Harris recurrence of the CIR process and its applications. Commun. Stoch. Anal. 7:409–424MathSciNetGoogle Scholar
  22. Jones CS (2003) Nonlinear mean reversion in the short-term interest rate. Rev. Financial Stud. 16:793–843CrossRefGoogle Scholar
  23. Jones MC, Marron JS, Sheather SJ (1996) A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91:401–407MathSciNetCrossRefMATHGoogle Scholar
  24. Moloche G (2001) Local nonparametric estimation of scalar diffusions. Preprint, Munich Personal RePEc ArchiveGoogle Scholar
  25. Podolskij M, Vetter M (2009) Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15:634–658MathSciNetCrossRefMATHGoogle Scholar
  26. Schmisser E (2011) Non-parametric drift estimation for diffusions from noisy data. Statist. Decisions 28:119–150MathSciNetCrossRefMATHGoogle Scholar
  27. Schmisser E (2013) Penalized nonparametric drift estimation for a multidimensional diffusion process. Statistics 47:61–84MathSciNetCrossRefMATHGoogle Scholar
  28. Schmisser E (2014) Non-parametric adaptive estimation of the drift for a jump diffusion process. Stochastic Process. Appl. 124:883–914MathSciNetCrossRefMATHGoogle Scholar
  29. Stanton R (1997) A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 52:1973–2002CrossRefGoogle Scholar
  30. Stone M (1974) Cross-validatory choice and assessment of statistical predictions, with discussion. J. Roy. Statist. Soc. Ser. B 36:111–147MathSciNetMATHGoogle Scholar
  31. Strauch C (2015) Sharp adaptive drift estimation for ergodic diffusions: the multivariate case. Stochastic Process. Appl. 125:2562–2602MathSciNetCrossRefMATHGoogle Scholar
  32. Strauch C (2016) Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions. Probab. Theory Related Fields 164:361–400MathSciNetCrossRefMATHGoogle Scholar
  33. Zhang L, Mykland PA, Aït-Sahalia Y (2005) A tale of two time scales: determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100:1394–1411MathSciNetCrossRefMATHGoogle Scholar
  34. Zhou B (1996) High-frequency data and volatility in foreign-exchange rates. J. Bus. Econom. Statist. 14:45–52Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of ChicagoChicagoUSA
  2. 2.Statistics DepartmentUniversity of British ColumbiaVancouverCanada
  3. 3.Mathematical InstituteUniversity of CologneCologneGermany

Personalised recommendations