Estimating the Block-Diagonal Idiosyncratic Covariance in High-Dimensional Factor Models

Factor models are often used to infer lower-dimensional correlation structures in data, especially when the number of variables grows close to or beyond the number of data points. The data covariance under a factor model structure is a combination of a low-rank component due to common factors and a diagonal or sparse idiosyncratic component. In this paper we consider the estimation of the idiosyncratic component under the assumption of grouped variables, which result in a block-diagonal matrix. We propose a shrinkage approach which ensures the positive definiteness of the estimated matrix, using either known group structures or clustering algorithms to determine them. The proposed methods are tested in a portfolio optimization scenario using simulations and historical data. The results show that the cluster based estimators yield improved performance in terms of out-of-sample portfolio variance, as well as remarkable stability in terms of resilience to the error in the estimated number of latent factors.