An Econometric Analysis of Modulated Realised Covariance, Regression and Correlation in Noisy Diffusion Models

Abstract

This paper introduces a new estimator to measure the ex-post covariation between high-frequency financial time series under market microstructure noise. We provide an asymptotic limit theory (including feasible central limit theorems) for standard methods such as regression, correlation analysis and covariance, for which we obtain the optimal rate of convergence. We demonstrate some positive semidefinite estimators of the covariation and construct a positive semidefinite estimator of the conditional covariance matrix in the central limit theorem. Furthermore, we indicate how the assumptions on the noise process can be relaxed and how our method can be applied to non-synchronous observations.We also present an empirical study of how high-frequency correlations, regressions and covariances change through time. * We thank Neil Shephard and Matthias Vetter for valuable comments. The first author gratefully acknowledges financial support from the Rhodes Trust and Oxford-Man Institute of Quantitative Finance. The second author is grateful for financial support from CREATES funded by the Danish National Research Foundation.

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

“…where a B (g, θ ) = θ 2 ψ 2 2 , a M (g, θ ) = ψ 1 ψ 2 and a N (g, θ ) = ψ 2 1 θ 2 (the proof of this result is achieved by using arguments alike the ones presented in the proof of Theorem 1; see Kinnebrock and Podolskij, 2008, for further details).…”

Efficient Covariance Estimation for Asynchronous Noisy High‐Frequency Data

“…The noise variances can be estimated using the realized variances (see e.g. Zhang et al, 2005) and a consistent estimator for (1 + 2 s )( X s Y s ) 2 dt can be found in Kinnebrock & Podolskij (2008) and Barndorff-Nielsen et al (2008b) where the asymptotic variance of the kernel estimator is given.…”

“…The problem of estimating the quadratic covariation from asynchronous data without noise has been solved by Hayashi & Yoshida (2005) and there are as well estimators developed in the recent literature that solve the problem of noisy but synchronous data; see e.g. Kinnebrock & Podolskij (2008) and Bandi & Russel (2005).…”

“…Their estimator has an N 1/5 rate of convergence. Kinnebrock & Podolskij (2008) developed an estimator for the synchronous multivariate setting based on the pre-average approach that is rate-optimal and also provides a positive semi-definite version with N 1/5 rate.In section 2, we introduce the model and our basic notation. There we also give an outline of the remaining sections of the article and present our two main results concerning the asymptotics of our generalized multi-scale estimator for the quadratic covariation and LAN.…”

“…Their estimator has an N 1/5 rate of convergence. Kinnebrock & Podolskij (2008) developed an estimator for the synchronous multivariate setting based on the pre-average approach that is rate-optimal and also provides a positive semi-definite version with N 1/5 rate.…”