Nonlinear shrinkage estimation of large integrated covariance matrices

Integrated covariance matrices arise in intra-day models of asset returns, which allow volatility to change across the trading day. When the number of assets is large, the natural estimator of such a matrix suffers from bias, contributed from extreme eigenvalues. We introduce a novel nonlinear shrinkage estimator for the integrated covariance matrix which shrinks the extreme eigenvalues of a realized covariance matrix back to an acceptable level, and enjoys a certain asymptotic efficiency when the number of assets is of the same order as the number of data points. Novel maximum exposure and actual risk bounds are derived when our estimator is used in constructing the minimum variance portfolio. Compared to other methods, our estimator performs favorably in both simulations and a real data analysis.