  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.
