High-Dimensional Structured Covariance Matrix Estimation with Financial Applications

Abstract

This thesis deals with high dimensional statistical inference, and more speci-

cally 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.

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 the sparsity modeling and the factor structure. In nancial applications, the residual

covariance represents idiosyncratic risk that can be diversied 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.