IRMACS Visiting Scholar Lecture Series: "Estimation of High Dimensional Covariance Matrix"

Abstract

High dimensionality comparable to the sample size is a common feature in portfolio allocation, risk management, genetic network and climatology. In this talk, we first use a multi-factor model to reduce the dimensionality and to estimate the covariance matrix for portfolio allocation and risk assessment. The impacts of dimensionality on the estimation of covariance matrix and its inverse are examined. We identify the situations under which the factor approach can gain substantially the performance and the cases where the gains are only marginal, in comparison with the sample covariance matrix. Furthermore, the impacts of the covariance matrix estimation on portfolio allocation and risk management are studied. Viable covariance modeling and sparse and robust portfolio allocations are recommended based on our mathematical results. In other class of problems such as genetic network or climatology, sparsity of the covariance matrix or its inverse arises naturally. We then estimate high-dimensional covariance matrices using the penalized likelihood method to explore the sparsity. New algorithms are proposed. Optimal rates of convergence, sparsistency, and asymptotic normality are established. Our theoretical results are verified by simulation studies and illustrated by several applications.