Abstract

Large dimensionality comparable to the sample size is a common feature as 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 management. 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. In other class of problems such as genetic network or climatology, sparsity of the covariance matrix or its inverse arises natural. We then estimate the large covariance matrix estimation by exploiting its sparsity using the penalized likelihood method. Sampling property is established and new algorithms are proposed.