Covariance Model with General Linear Structure and Divergent Parameters

TL;DR

This paper introduces a flexible covariance modeling approach that connects the covariance matrix to linear combinations of weight matrices, allowing for divergence in parameters and providing estimators with proven asymptotic properties, validated through simulations and stock market data.

Contribution

It proposes the covariance model with general linear structure (CMGL), extending covariance estimation to diverging parameters without distribution assumptions, and develops new estimators and model selection criteria.

Findings

QMLE estimators are asymptotically consistent.

EBIC effectively selects relevant weight matrices.

OLS estimator offers computational efficiency with solid theoretical backing.

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 US stock market.