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Bipower-type estimation in a noisy diffusion setting

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Author Info

  • Mark Podolskij
  • Mathias Vetter

    ()
    (School of Economics and Management, University of Aarhus, Denmark and CREATES)

Abstract

We consider a new class of estimators for volatility functionals in the setting of frequently observed Itô diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general Itô semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2008-25.

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Length: 42
Date of creation: 26 May 2008
Date of revision:
Handle: RePEc:aah:create:2008-25

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: Bipower Variation; Central Limit Theorem; High-Frequency Data; Microstructure Noise; Quadratic Variation; Semimartingale Theory; Test for Jumps;

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References

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  1. Ole E Barndorff-Nielsen & Peter Hansen & Asger Lunde & Neil Shephard, 2006. "Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise," OFRC Working Papers Series, Oxford Financial Research Centre 2006fe05, Oxford Financial Research Centre.
  2. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
  3. Jean Jacod & Yingying Li & Per A. Mykland & Mark Podolskij & Mathias Vetter, 2007. "Microstructure Noise in the Continuous Case: The Pre-Averaging Approach - JLMPV-9," CREATES Research Papers, School of Economics and Management, University of Aarhus 2007-43, School of Economics and Management, University of Aarhus.
  4. Ole E. Barndorff-Nielsen & Shephard, 2002. "Econometric analysis of realized volatility and its use in estimating stochastic volatility models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 64(2), pages 253-280.
  5. Ole Barndorff-Nielsen & Svend Erik Graversen & Jean Jacod & Mark Podolskij & Neil Shephard, 2004. "A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales," Economics Papers, Economics Group, Nuffield College, University of Oxford 2004-W29, Economics Group, Nuffield College, University of Oxford.
  6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  7. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 2(1), pages 1-37.
  8. Vetter, Mathias & Podolskij, Mark, 2006. "Estimation of Volatility Functionals in the Simultaneous Presence of Microstructure Noise and Jumps," Technical Reports, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen 2006,51, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  9. Jacod, Jean & Li, Yingying & Mykland, Per A. & Podolskij, Mark & Vetter, Mathias, 2007. "Microstructure noise in the continuous case: the pre-averaging approach," Technical Reports, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen 2007,41, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  10. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, Cambridge University Press, vol. 12(04), pages 627-627, November.
  11. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, Elsevier, vol. 5(2), pages 177-188, November.
  12. Jacod, Jean, 2008. "Asymptotic properties of realized power variations and related functionals of semimartingales," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 118(4), pages 517-559, April.
  13. Chan, K C, et al, 1992. " An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, American Finance Association, vol. 47(3), pages 1209-27, July.
  14. Lan Zhang & Per A. Mykland & Yacine Ait-Sahalia, 2003. "A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High Frequency Data," NBER Working Papers 10111, National Bureau of Economic Research, Inc.
  15. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1980. " An Analysis of Variable Rate Loan Contracts," Journal of Finance, American Finance Association, American Finance Association, vol. 35(2), pages 389-403, May.
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Citations

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Cited by:
  1. Nikolaus Hautsch & Mark Podolskij, 2010. "Pre-Averaging Based Estimation of Quadratic Variation in the Presence of Noise and Jumps: Theory, Implementation, and Empirical Evidence," CREATES Research Papers, School of Economics and Management, University of Aarhus 2010-29, School of Economics and Management, University of Aarhus.
  2. Christensen, Kim & Podolskij, Mark & Vetter, Mathias, 2013. "On covariation estimation for multivariate continuous Itô semimartingales with noise in non-synchronous observation schemes," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 120(C), pages 59-84.
  3. Koike, Yuta, 2014. "Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 124(8), pages 2699-2753.
  4. Jean Jacod & Mark Podolskij & Mathias Vetter, 2008. "Intertemporal Asset Allocation with Habit Formation in Preferences: An Approximate Analytical Solution," CREATES Research Papers, School of Economics and Management, University of Aarhus 2008-61, School of Economics and Management, University of Aarhus.
  5. M. Podolskij & D. Ziggel, 2010. "New tests for jumps in semimartingale models," Statistical Inference for Stochastic Processes, Springer, Springer, vol. 13(1), pages 15-41, April.
  6. Valentina Corradi & Norman Swanson & Walter Distaso, 2006. "Predictive Inference for Integrated Volatility," Departmental Working Papers, Rutgers University, Department of Economics 200616, Rutgers University, Department of Economics.

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