  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.
