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.

内容简介:该论文系统研究了在观测到的高维高频数据具有异步性（asynchronicity）以及存在相依的随机测量误差的情形下，如何对随机测量误差的协方差矩阵进行估计。具体而言，该论文通过局部化和适当的截断给出了一个关于高维高频数据中随机测量误差协方差矩阵的估计，并在两种常用的损失函数下研究了该估计的收敛速度以及收敛速度达到极小极大最优（minimax optimal）的情形。考虑到该协方差矩阵中的元素在实际数据中可能很小，该论文也对所提出的估计进行了二阶理论分析，给出了该估计的渐近偏差，提出了一个基于偏差校正的估计，以提升有限样本下的估计效果。同时，为了研究所提方法的鲁棒性，该论文还考虑了观测数据带有跳跃项的情形，理论结果表明该估计在此情形下仍然具有相同的收敛速度。

关键词：High-dimensional covariance matrix; High-frequency data analysis; Measurement error; Minimax optimality; Thresholding

该文2022年9月在线发表于计量经济学国际顶尖刊物《Journal of Econometrics》。该期刊为经管院A类期刊。

论文链接：https://www.sciencedirect.com/science/article/pii/S0304407622001543