**No.**
1067

**Title:**

- Nonsynchronous covariance estimator and limit theorem II

**Author(s): **

- Takaki Hayashi (Keio University);

Nakahiro Yoshida (University of Tokyo)

**Key words: **

- discreate sampling; high-frequency data; martingale central limit theorem; nonsynchronicity; quadratic variation; realized volatility; stable convergence; semimartingale.

**Abstract: **

- An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales (Hayashi and Yoshida (2005b), Hayashi and Yoshida (2006)) is studied. In the setup, two continuous semimartingales are sampled at stopping times, in a nonsynchronous manner. The nonsynchronous covariation is a consistent estimator for the 'true' quadratic covariation of the semimartingales, as the mesh size of the sampling intervals shrinks to zero. In particular, we deal with the case when the limiting variation process of the normalized "approximation error" is random, which leads to the convergence to mixed normality, or "conditional" Gaussian martingale. A class of consistent estimators for the asymptotic variation process is proposed based on kernels, which will be useful from a viewpoint of statistical inferences. An example is presented.