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Edgeworth expansion for the pre-averaging estimator

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  • Podolskij, Mark
  • Veliyev, Bezirgen
  • Yoshida, Nakahiro

Abstract

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.

Suggested Citation

  • Podolskij, Mark & Veliyev, Bezirgen & Yoshida, Nakahiro, 2017. "Edgeworth expansion for the pre-averaging estimator," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3558-3595.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:11:p:3558-3595
    DOI: 10.1016/j.spa.2017.03.001
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    1. Sílvia Gonçalves & Ulrich Hounyo & Nour Meddahi, 2014. "Bootstrap Inference for Pre-averaged Realized Volatility based on Nonoverlapping Returns," Journal of Financial Econometrics, Oxford University Press, vol. 12(4), pages 679-707.
    2. Jacod, Jean & Li, Yingying & Mykland, Per A. & Podolskij, Mark & Vetter, Mathias, 2009. "Microstructure noise in the continuous case: The pre-averaging approach," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2249-2276, July.
    3. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
    4. Ole E. Barndorff-Nielsen & Peter Reinhard Hansen & Asger Lunde & Neil Shephard, 2008. "Designing Realized Kernels to Measure the ex post Variation of Equity Prices in the Presence of Noise," Econometrica, Econometric Society, vol. 76(6), pages 1481-1536, November.
    5. Nikolaus Hautsch & Mark Podolskij, 2013. "Preaveraging-Based Estimation of Quadratic Variation in the Presence of Noise and Jumps: Theory, Implementation, and Empirical Evidence," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(2), pages 165-183, April.
    6. Ulrich Hounyo & Bezirgen Veliyev, 2016. "Validity of Edgeworth expansions for realized volatility estimators," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
    7. Mark Podolskij & Nakahiro Yoshida, 2013. "Edgeworth expansion for functionals of continuous diffusion processes," CREATES Research Papers 2013-33, Department of Economics and Business Economics, Aarhus University.
    8. Yoshida, Nakahiro, 2013. "Martingale expansion in mixed normal limit," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 887-933.
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    Cited by:

    1. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    2. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    3. Tudor, Ciprian A. & Yoshida, Nakahiro, 2023. "High order asymptotic expansion for Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 443-492.
    4. Yamagishi, Hayate & Yoshida, Nakahiro, 2023. "Order estimate of functionals related to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 490-543.
    5. Ciprian A. Tudor & Nakahiro Yoshida, 2020. "Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 435-463, July.

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    More about this item

    Keywords

    Diffusion processes; Edgeworth expansion; high frequency observations; quadratic variation; pre-averaging;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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