Some topics in analyzing high-dimensional covariance matrices

Abstract

This thesis considers two problems related to high-dimensional covariance matrices, namely, covariance matrix estimation and multivariate volatility modeling. Covariance matrix estimation is important in many statistical methods and applications. For example, it is applied in asset allocation, classification of human tumors based on gene expression arrays, and many others. Sample covariance matrix is frequently used as an estimator of the population covariance matrix. However, sample covariance matrix becomes poor and unstable with the increase in the dimensions of the data vectors. A typical problem is that the eigenvalues of the population covariance matrix become distorted. This thesis proposes a method for resolving this problem, namely, an efficient estimation method that imposes a new likelihood penalty on the covariance matrix. The proposed estimator is guaranteed to be non-negative definite and its estimation algorithm is fast and efficient. The proposed method is compared with several existing methods via simulation and empirical studies.

Another interesting topic in analyzing high-dimensional covariance matrices is multivariate volatility modeling, which is crucial in option pricing, portfolio risk forecasting, and risk measurement and management. The increasing availability of intra-day trading data has drawn widespread attention to the study of daily realized covariance matrices constructed from high-frequency data. The existing methods for reducing the dimensions of a sample of covariance matrices include matrix-based methods such as matrix factor (MFA) model and common component analysis (CCA), and vector-based method such as principal component analysis (PCA). The relationship and differences in performance between these two classes of methods have yet to be explored in the literature. In this thesis, I fill this research gap by comparing the methods theoretically and empirically.

Note that the factor loading matrix in the MFA model is constant over time, and hence it may not work well in forecasting high-dimensional covariance matrices. To resolve this problem, a dynamic matrix factor (DMF) model is proposed to allow the loading matrix to vary over time. The DMF model assumes that the realized covariance matrix follows a central Wishart distribution with the conditional expectation of the realized covariance matrix being decomposed into a loading matrix and a diagonal matrix via spectral decomposition. In the DMF model, a loading-driven process with the scalar BEKK model is used to capture the dynamics of the loading matrix, and each diagonal term is modeled by a separate GARCH(1,1) model. The forecasts of the realized covariance matrices are guaranteed to be positive definite. To maintain a parsimonious model structure, the DMF model is extended to the case of high-dimensional covariance matrix. Finally, the proposed DMF models are applied to several real-world data sets.