On covariation estimation for multivariate continuous It\^o semimartingales with noise in non-synchronous observation schemes

TL;DR

This paper introduces a new estimator for the covariation matrix of multivariate continuous Itô semimartingales observed with noise at non-synchronous times, which is robust and does not require synchronization.

Contribution

The paper proposes a Hayashi-Yoshida type estimator that combines local averages and is robust to noise dependence and non-synchronous observations, without needing synchronization.

Findings

The estimator is consistent and asymptotically normal.

Simulation results demonstrate good finite sample performance.

The method is robust to certain noise dependence structures.

Abstract

This paper presents a Hayashi-Yoshida type estimator for the covariation matrix of continuous It\^o semimartingales observed with noise. The coordinates of the multivariate process are assumed to be observed at highly frequent non-synchronous points. The estimator of the covariation matrix is designed via a certain combination of the local averages and the Hayashi-Yoshida estimator. Our method does not require any synchronization of the observation scheme (as e.g. previous tick method or refreshing time method) and it is robust to some dependence structure of the noise process. We show the associated central limit theorem for the proposed estimator and provide a feasible asymptotic result. Our proofs are based on a blocking technique and a stable convergence theorem for semimartingales. Finally, we show simulation results for the proposed estimator to illustrate its finite sample properties.