Limit theorems for the pre-averaged Hayashi-Yoshida estimator with random sampling

TL;DR

This paper establishes the theoretical properties of a modified pre-averaged Hayashi-Yoshida estimator for estimating integrated covariance from high-frequency, nonsynchronously observed, noisy, and potentially dependent data, proving its consistency and asymptotic normality.

Contribution

It introduces a new version of the estimator that accounts for noise and dependence in sampling times, providing rigorous asymptotic results.

Findings

Estimator is consistent and asymptotically mixed normal.

Achieves the optimal rate of convergence.

Handles noise and dependent sampling times effectively.

Abstract

We will focus on estimating the integrated covariance of two diffusion processes observed in a nonsynchronous manner. The observation data is contaminated by some noise, which is possibly correlated with the returns of the diffusion processes, while the sampling times also possibly depend on the observed processes. In a high-frequency setting, we consider a modified version of the pre-averaged Hayashi-Yoshida estimator, and we show that such a kind of estimators has the consistency and the asymptotic mixed normality, and attains the optimal rate of convergence.