Central limit theorems for pre-averaging covariance estimators under endogenous sampling times

TL;DR

This paper proves a central limit theorem for pre-averaged covariance estimators in a setting with endogenous sampling times, showing their robustness compared to realized volatility.

Contribution

It establishes a central limit theorem for the pre-averaged Hayashi-Yoshida estimator under endogenous sampling, highlighting its robustness and contrasting it with realized volatility.

Findings

Endogeneity does not affect the asymptotic distribution of the pre-averaged estimator.

The pre-averaging technique uniquely maintains its properties under endogenous sampling.

Central limit theorems are also established for the modulated realized covariance.

Abstract

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.