Covariance Model with General Linear Structure and Divergent Parameters

Abstract

For estimating the large covariance matrix with a limited sample size, we propose the covariance model with general linear structure (CMGL) by employing the general link function to connect the covariance of the continuous response vector to a linear combination of weight matrices. Without assuming the distribution of responses, and allowing the number of parameters associated with weight matrices to diverge, we obtain the quasi-maximum likelihood estimators (QMLE) of parameters and show their asymptotic properties. In addition, an extended Bayesian information criteria (EBIC) is proposed to select relevant weight matrices, and the consistency of EBIC is demonstrated. Under the identity link function, we introduce the ordinary least squares estimator (OLS) that has the closed form. Hence, its computational burden is reduced compared to QMLE, and the theoretical properties of OLS are also investigated. To assess the adequacy of the link function, we further propose the quasi-likelihood ratio test and obtain its limiting distribution. Simulation studies are presented to assess the performance of the proposed methods, and the usefulness of generalized covariance models is illustrated by an analysis of the U.S. stock market.

KEYWORD:

• Covariance models with general linear structure
• Diverging parameters
• Extended Bayesian information criteria
• Linear covariance structure
• Quasi-likelihood ratio test

Supplementary Materials

The supplementary material consists of nine sections. Section S.1 presents a detailed illustration of G and its related derivatives. Section S.2 introduces an example that satisfies Conditions 2–5. Section S.3 presents twelve useful lemmas; Sections S.4–S.6 demonstrate Theorems 3–5, respectively; Section S.7 extends our model to accommodate $n\ge 1$; Section S.8 introduces the proofs of Theorems 7 and 8 that are proposed in Section S.7; Section S.9 presents the simulation results for the mixture normal and the standardized exponential distributions.

Acknowledgments

The authors are grateful to the editor, associate editor, and anonymous referees for their insightful comments and constructive suggestions.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Xinyan Fan’s research was supported by the National Natural Science Foundation of China (NSFC, 12201626), and the Public Computing Cloud, Renmin University of China. Wei Lan’s research was supported by the National Natural Science Foundation of China (NSFC, 71991472, 12171395, 11931014), the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics, and the Fundamental Research Funds for the Central Universities (JBK1806002). Tao Zou’s research was supported by ANU College of Business and Economics Early Career Researcher Grant, the RSFAS Cross Disciplinary Grant.