High-Dimensional Dynamic Covariance Matrices With Homogeneous Structure

High-dimensional covariance matrices appear in many disciplines. Much literature has devoted to the research in high-dimensional constant covariance matrices. However, constant covariance matrices are not sufficient in applications, for example, in portfolio allocation, dynamic covariance matrices would be more appropriate. As argued in this article, there are two difficulties in the introduction of dynamic structures into covariance matrices: (1) simply assuming each entry of a covariance matrix is a function of time to introduce the dynamic needed would not work; (2) there is a risk of having too many unknowns to estimate due to the high dimensionality. In this article, we propose a dynamic structure embedded with a homogeneous structure. We will demonstrate the proposed dynamic structure makes more sense in applications and avoids, in the meantime, too many unknown parameters/functions to estimate, due to the embedded homogeneous structure. An estimation procedure is also proposed to estimate the proposed high-dimensional dynamic covariance matrices, and asymptotic properties are established to justify the proposed estimation procedure. Intensive simulation studies show the proposed estimation procedure works very well when the sample size is finite. Finally, we apply the proposed high-dimensional dynamic covariance matrices to portfolio allocation. It is interesting to see the resulting portfolio yields much better returns than some commonly used ones.