Author: Jinyuan Chang, Qiao Hu, Cheng Liu, and Cheng Yong Tang

Abstract: We consider high-dimensional measurement errors with high-frequency data. Our objective is on recovering the high-dimensional cross-sectionalcovariance matrixof the random errors withoptimality. In this problem, not all components of the random vector are observed at the same time and the measurement errors are latent variables, leading to major challenges besides high data dimensionality. We propose a new covariance matrix estimator in this context with appropriate localization and thresholding, and then conduct a series of comprehensive theoretical investigations of the proposed estimator. By developing a new technical device integrating the high-frequency data feature with the conventional notion ofα-mixing, our analysis successfully accommodates the challenging serial dependence in the measurement errors. Our theoretical analysis establishes theminimaxoptimal convergence rates associated with two commonly used loss functions; and we demonstrate with concrete cases when the proposed localized estimator with thresholding achieves the minimax optimal convergence rates. Considering that the variances and covariances can be small in reality, we conduct a second-order theoretical analysis that further disentangles the dominating bias in the estimator. A bias-corrected estimator is then proposed to ensure its practical finite sample performance. We also extensively analyze our estimator in the setting with jumps, and show that its performance is reasonably robust. We illustrate the promising empirical performance of the proposed estimator with extensive simulation studies and a real data analysis.

Key words: High-dimensional covariance matrix; High-frequency data analysis; Measurement error; Minimax optimality; Thresholding

This paper was published online in September 2022 in the Journal of Econometrics, A class Journal of the School of Economics and Management, Wuhan University

Paper link: https://www.sciencedirect.com/science/article/pii/S0304407622001543