High-Dimensional Structured Covariance Matrix Estimation with Financial Applications

This thesis deals with high dimensional statistical inference, and more specifically with uncovering low dimensional structures in high dimensional systems. It focuses on large covariance and precision matrix estimation under various complexity constraints, such as low rank and conditional sparsity properties. I have also demonstrated their applications in portfolio allocation and risk minimization. In recent years, high-dimensional settings have become one of the major focuses of statistics research. The interest has been driven by applications in finance, genetics, image processing, etc. and technological advances have made large data sets widely available for examination. In a data rich environment, the number of parametes can diverge at a rate faster than that of the sample size. A natural way of dealing with this challenge is establishing low dimensional patterns in the data, such as low-rank matrices and sparsity. The low-rank approach assumes that there is an underlying lower-dimensional subspace that drives the random processes. Sparsity implies that a large number of the coefficients to be estimated are equal to zero, and the number of non-zero entries is restricted to grow slowly. My thesis is on high dimensional covariance matrix estimation with a conditional sparsity structure and fast diverging eigenvalues. I consider the high dimensional approximate factor model, in which the number of units grows possibly exponentially fast with the sample size. Classical methods of estimating the covariance matrices in factor models are based on the cross-sectional independence assumption among the idiosyncratic components. This assumption, however, is restrictive in practical applications. For example, returns depend on equity market risks, housing prices depend on economic health. By assuming the error covariance matrix to be sparse, I allow the presence of the cross-sectional correlation, and combine the merits of both iii the sparsity modeling and the factor structure. In financial applications, the residual covariance represents idiosyncratic risk that can be diversified away, and so makes a smaller order contribution to portfolio risk, but in practice it can be important. I study the impact of weakly dependent data with strong mixing conditions on estimation, and obtain asymptotically nonsingular estimators for the covariance matrices using various thresholding techniques. It is shown that the estimated covariance matrices are consistent under various norms. Both observable and unobservable factor cases are considered. The uniform rates of convergence for the factor loadings and the unobservable factors are also derived. This approach is simple and optimization free and it uses the data only through the sample covariance matrix. The results of the thesis are separated into two chapters. Chapter 1 is based on the assumption that direct observations of the common factors are available. The true number of factors is also given. We take the classical definition of sparsity for the error covariance matrix, namely that many of its off-diagonal elements are zero. We estimate the sparse covariance using the adaptive thresholding technique as in Cai and Liu (2011), taking into account the fact that the true idiosyncratic components are unknown. Chapter 2 explores a much more general setup. We introduce the Principal Orthogonal complEment Thresholding (POET) method for covariance estimation through principal component analysis of the data. We provide mathematical insights for the connection between this method and the approximate factor model with sparsity for high-dimensional data. The common factors are unobservable, and their number is also unknown. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. We use a generalized notion of sparsity and a broader thresholding technique, namely the generazlied shrinkage