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Time endogeneity and an optimal weight function in pre-averaging covariance estimation

Abstract

We establish a central limit theorem for a class of pre-averaging covariance estimators in a general endogenous time setting. In particular, we show that the time endogeneity has no impact on the asymptotic distribution if certain functionals of observation times are asymptotically well-defined. This contrasts with the case of the realized volatility in a pure diffusion setting. We also discuss an optimal choice of the weight function in the pre-averaging.

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Notes

  1. The preliminary version Koike (2013) of this paper focuses on the pre-averaged Hayashi-Yoshida estimator, which is another covariance estimator introduced in Christensen et al. (2010).

  2. We set \(\sum _{i=p}^q\equiv 0\) if \(p>q\) by convention.

  3. This point can be solved by pre-synchronizing the data similarly to our case, i.e. consider \(\overline{Y}^k_{i}\) instead of \(\overline{Y}^k_{t^k_i}\); see Koike (2014) for details. See also Sect. 6.3 of Bibinger (2012) where other advantages of such a procedure are discussed for the case of the subsampling approach.

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Acknowledgments

The author thanks the Editors and an anonymous referee for their careful reading and helpful comments. In particular, the author strongly appreciates the referee for having pointed out errors in the statement of Proposition 5.1 and the proof of Proposition 6.1 contained in earlier versions of the paper. The author is also grateful to Teppei Ogihara who pointed out a problem on the mathematical construction of the noise process in a previous version of this paper. This work was partly supported by Grant-in-Aid for JSPS Fellows, the Program for Leading Graduate Schools, MEXT, Japan and CREST JST.

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Correspondence to Yuta Koike.

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Koike, Y. Time endogeneity and an optimal weight function in pre-averaging covariance estimation. Stat Inference Stoch Process 20, 15–56 (2017). https://doi.org/10.1007/s11203-016-9135-3

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  • DOI: https://doi.org/10.1007/s11203-016-9135-3

Keywords

  • Central limit theorem
  • Jumps
  • Market microstructure noise
  • Non-synchronous observations
  • Pre-averaging
  • Time endogeneity