Penalized Sparse Covariance Regression with High Dimensional Covariates

Covariance regression offers an effective way to model the large covariance matrix with the auxiliary similarity matrices. In this work, we propose a sparse covariance regression (SCR) approach to handle the potentially high-dimensional predictors (i.e., similarity matrices). Specifically, we use the penalization method to identify the informative predictors and estimate their associated coefficients simultaneously. We first investigate the Lasso estimator and subsequently consider the folded concave penalized estimation methods (e.g., SCAD and MCP). However, the theoretical analysis of the existing penalization methods is primarily based on i.i.d. data, which is not directly applicable to our scenario. To address this difficulty, we establish the non-asymptotic error bounds by exploiting the spectral properties of the covariance matrix and similarity matrices. Then, we derive the estimation error bound for the Lasso estimator and establish the desirable oracle property of the folded concave penalized estimator. Extensive simulation studies are conducted to corroborate our theoretical results. We also illustrate the usefulness of the proposed method by applying it to a Chinese stock market dataset.