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Nonsynchronous covariation process and limit theorems

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  • Hayashi, Takaki
  • Yoshida, Nakahiro

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.

Suggested Citation

  • Hayashi, Takaki & Yoshida, Nakahiro, 2011. "Nonsynchronous covariation process and limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2416-2454, October.
  • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2416-2454
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    References listed on IDEAS

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    1. Takaki Hayashi & Nakahiro Yoshida, 2008. "Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 367-406, June.
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    5. Toshiya Hoshikawa & Keiji Nagai & Taro Kanatani & Yoshihiko Nishiyama, 2008. "Nonparametric Estimation Methods of Integrated Multivariate Volatilities," Econometric Reviews, Taylor & Francis Journals, vol. 27(1-3), pages 112-138.
    6. Masato Ubukata & Kosuke Oya, 2008. "A Test for Dependence and Covariance Estimator of Market Microstructure Noise," Discussion Papers in Economics and Business 07-03-Rev.2, Osaka University, Graduate School of Economics.
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