Estimation of Large Dynamic Covariance Matrices:A Selective Review

A personal review of some recent developments on estimating large dynamic covariance matrices whose entries are allowed to change over time is provided. The underlying covariance matrices are assumed to satisfy structural assumptions such as GARCH, approximate sparsity and conditional sparsity. Initially the review considers extensions of the classic GARCH model to multivariate and high-dimensional time series settings, and then focuses on some data-driven non- and semi-parametric models and estimation approaches for large covariance matrices which evolve smoothly over time or with some conditioning variables. Detection of multiple structural breaks in large covariance structures is also reviewed. Finally some relevant future directions are discussed.