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Nonsynchronous covariation process and limit theorems

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  • Hayashi, Takaki
  • Yoshida, Nakahiro
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    Abstract

    An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 121 (2011)
    Issue (Month): 10 (October)
    Pages: 2416-2454

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    Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2416-2454

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    Related research

    Keywords: Discrete sampling High-frequency data Martingale central limit theorem Nonsynchronicity Quadratic variation Realized volatility Stable convergence Semimartingale;

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    Cited by:
    1. Markus Bibinger & Mathias Vetter, 2013. "Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps," SFB 649 Discussion Papers SFB649DP2013-029, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    2. Djellout, Hacène & Samoura, Yacouba, 2014. "Large and moderate deviations of realized covolatility," Statistics & Probability Letters, Elsevier, vol. 86(C), pages 30-37.
    3. Markus Bibinger & Nikolaus Hautsch & Peter Malec & Markus Reiss, 2013. "Estimating the Quadratic Covariation Matrix from Noisy Observations: Local Method of Moments and Efficiency," SFB 649 Discussion Papers SFB649DP2013-017, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Koike, Yuta, 2014. "Limit theorems for the pre-averaged Hayashi–Yoshida estimator with random sampling," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2699-2753.
    5. Bibinger, Markus, 2012. "An estimator for the quadratic covariation of asynchronously observed Itô processes with noise: Asymptotic distribution theory," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2411-2453.

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