When estimating high-frequency covariance (quadratic covariation) of two arbitrary assets observed asynchronously, simple assumptions  are usually imposed on the relationship between the prices process and the observation times. In this paper, we introduce a very general endogenous two dimensional nonparametric model. Because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes, it is called the hitting boundary process with time process (HBT) model. We establish a central limit theorem for the Hayashi-Yoshida  estimator under HBT in the case where prices process follows a continuous Ito process : we obtain an asymptotic bias. We provide an estimator of the latter as well as a bias-corrected  estimator of the high-frequency covariance. In addition, we give a consistent estimator of the associated standard error.
When estimating high-frequency covariance (quadratic covariation) of two arbitrary assets observed asynchronously, simple assumptions , such as independence, are usually imposed on the relationship between the prices process and the observation times. In this paper, we introduce a general endogenous two-dimensional nonparametric model. Because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes, it is called the hitting boundary process with time process (HBT) model. We establish a central limit theorem for the Hayashi-Yoshida (HY) estimator under HBT in the case where the price process and the observation price process follow a continuous Ito process . We obtain an asymptotic bias. We provide an estimator of the latter as well as a bias-corrected HY estimator of the high-frequency covariance. In addition, we give a consistent estimator of the associated standard error.
A
, such as independence,
R
very general endogenous two dimensional
general endogenous two-dimensional
A
(HY)
R
prices process follows
the price process and the observation price process follow
R
: we
. We
A
HY