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

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.