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Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps

Abstract

We consider estimation of the quadratic (co)variation of a semimartingale from discrete observations which are irregularly spaced under high-frequency asymptotics. In the univariate setting, results by Jacod for regularly spaced observations are generalized to the case of irregular observations. In the two-dimensional setup under non-synchronous observations, we derive a stable central limit theorem for the Hayashi–Yoshida estimator in the presence of jumps. We reveal how idiosyncratic and simultaneous jumps affect the asymptotic distribution. Observation times generated by Poisson processes are explicitly discussed.

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Acknowledgments

Markus Bibinger gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft via SFB 649 “Ökonomisches Risiko”, Humboldt-Universität zu Berlin. Mathias Vetter is thankful for financial support through the collaborative research center “Statistik nichtlinearer dynamischer Prozesse” (SFB 823) of the Deutsche Forschungsgemeinschaft. Both authors are grateful to an anonymous referee whose valuable comments helped improving the paper considerably.

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Bibinger, M., Vetter, M. Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps. Ann Inst Stat Math 67, 707–743 (2015). https://doi.org/10.1007/s10463-014-0473-x

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  • DOI: https://doi.org/10.1007/s10463-014-0473-x

Keywords

  • Asynchronous observations
  • Co-jumps
  • Statistics of semimartingales
  • Quadratic covariation