  This paper examines the usefulness of high frequency data in estimating the
covariance matrix for portfolio choice when the portfolio size is large. A
computationally convenient nonlinear shrinkage estimator for the integrated
covariance (ICV) matrix of financial assets is developed in two steps. The
eigenvectors of the ICV are first constructed from a designed time variation
adjusted realized covariance matrix of noise-free log-returns of relatively low
frequency data. Then the regularized eigenvalues of the ICV are estimated by
quasi-maximum likelihood based on high frequency data. The estimator is always
positive definite and its inverse is the estimator of the inverse of ICV. It
minimizes the limit of the out-of-sample variance of portfolio returns within
the class of rotation-equivalent estimators. It works when the number of
underlying assets is larger than the number of time series observations in each
asset and when the asset price follows a general stochastic process. Our
theoretical results are derived under the assumption that the number of assets
(p) and the sample size (n) satisfy p/n \to y >0 as n goes to infty . The
advantages of our proposed estimator are demonstrated using real data.
