Optimal covariance matrix estimation for high-dimensional noise in high-frequency data

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