Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency

TL;DR

This paper introduces an efficient, asymptotically optimal estimator for the integrated co-volatility matrix of multivariate continuous martingales, effectively handling noisy and nonsynchronous high-frequency data.

Contribution

It develops a local generalized method of moments estimator in the spectral domain that achieves asymptotic semi-parametric efficiency, with robustness to nonsynchronicity and improved performance under correlation.

Findings

Nonsynchronicity does not affect asymptotic results.

Efficiency gains are significant when correlations are high.

Simulations confirm good finite-sample performance.

Abstract

An efficient estimator is constructed for the quadratic covariation or integrated co-volatility matrix of a multivariate continuous martingale based on noisy and nonsynchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semi-parametric efficiency is established in the Cram\'{e}r-Rao sense. Main findings are that nonsynchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation. Simulations illustrate the finite-sample behaviour.