Integrated covariance matrix estimation for high-dimensional diffusion processes in the presence of microstructure noise

This article considers estimation of the integrated covariance (ICV) matrices of high-dimensional diffusion processes based on high-frequency data in the presence of microstructure noise. We adopt the pre-averaging approach to deal with microstructure noise, and establish the connection between the underlying ICV matrix and the pre-averaging estimator in terms of their limiting spectral distributions (LSDs). A key element of the argument is a result describing how the LSD of (true) sample covariance matrices depends on that of sample covariance matrices constructed from \emph{noisy} observations. This result enables one to make inferences about the covariance structure of underlying signals based on noisy observations. We further propose an alternative estimator, the pre-averaging time-variation adjusted realized covariance matrix, which possesses two desirable properties: it eliminates the impact of noise, and its LSD depends only on that of the targeting ICV through the standard Mar\v{c}enko-Pastur equation when the covolatility process satisfies certain structural conditions.