Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.

翻译：潜在因子模型中高维协方差矩阵的估计是许多领域尤其是金融学中的重要课题。随着金融资产数量的增长而估计窗口长度保持有限，常用的样本估计量会产生噪声估计，甚至无法保证正定性。在潜在因子模型假设下，协方差矩阵被分解为公共的低秩成分和满秩的特质成分。本文重点研究在时间序列存在分组结构（可能由行业、资产类别或国家等特定因素形成）假设下特质成分的估计问题。我们提出了一种广义方法，通过对残差序列进行聚类并对所得块应用收缩技术来估计块对角特质成分，从而确保正定性。基于不同聚类方法我们推导出两种估计量，并通过仿真与历史数据检验其性能。研究表明所提方法能提供可靠估计，且优于基于阈值方法的最先进估计量。