  We consider two continuous It\^o semimartingales observed with noise and
sampled at stopping times in a nonsynchronous manner. In this article we
establish a central limit theorem for the pre-averaged Hayashi-Yoshida
estimator of their integrated covariance in a general endogenous time setting.
In particular, we show that the time endogeneity has no impact on the
asymptotic distribution of the pre-averaged Hayashi-Yoshida estimator, which
contrasts the case for the realized volatility in a pure diffusion setting. We
also establish a central limit theorem for the modulated realized covariance,
which is another pre-averaging based integrated covariance estimator, and
demonstrate the above property seems to be a special feature of the
pre-averaging technique.
