Eigenvalue-regularized covariance matrix sstimators for high-dimensional data.

Covariance regularization is important when the dimension p of a covariance matrix is close to or even larger than the sample size n. This thesis concerns estimating large covariance matrix in both low and high frequency setting. First, we introduce an integration covariance matrix estimator which is a linear combination of a rotation-equivariant and a regularized covariance matrix estimator that assumed a specific structure for true covariance Σ0, under the practical scenario where one is not 100% certain of which regularization method to use. We estimate the weights in the linear combination and show that they asymptotically go to the true underlying weights. To generalize, we can put two regularized estimators into the linear combination, each assumes a specific structure for Σ0. Our estimated weights can then be shown to go to the true weights too, and if one regularized estimator is converging to Σ0 in the spectral norm, the corresponding weight then tends to 1 and others tend to 0 asymptotically. We demonstrate the performance of our estimator when compared to other state-of-the-art estimators through extensive simulation studies and a real data analysis. Next, in high-frequency setting with non-synchronous trading and contamination of microstructure noise, we propose a Nonparametrically Eigenvalue-Regularized Integrated coVariance matrix Estimator (NERIVE) which does not assume specific structures for the underlying integrated covariance matrix. We show that NERIVE is positive definite in probability, with extreme eigenvalues shrunk nonlinearly under the high dimensional framework p/n → c > 0. We also prove that in portfolio allocation, the minimum variance optimal weight vector constructed using NERIVE has maximum exposure and actual risk upper bounds of order p−1/2. The practical performance of NERIVE is illustrated by comparing to the usual two-scale realized covariance matrix as well as some other nonparametric alternatives using different simulation settings and a real data set. Last, another nonlinear shrinkage estimator of large integrated covariance matrix in high-frequency setting is explored, 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. In simulations and a real data analysis, our estimator performs favourably in comparison with other methods.