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For estimating the large covariance matrix with a limited sample size, wepropose the covariance model with general linear structure (CMGL) by employingthe general link function to connect the covariance of the continuous responsevector to a linear combination of weight matrices. Without assuming thedistribution of responses, and allowing the number of parameters associatedwith weight matrices to diverge, we obtain the quasi-maximum likelihoodestimators (QMLE) of parameters and show their asymptotic properties. Inaddition, an extended Bayesian information criteria (EBIC) is proposed toselect relevant weight matrices, and the consistency of EBIC is demonstrated.Under the identity link function, we introduce the ordinary least squaresestimator (OLS) that has the closed form. Hence, its computational burden isreduced compared to QMLE, and the theoretical properties of OLS are alsoinvestigated. To assess the adequacy of the link function, we further proposethe quasi-likelihood ratio test and obtain its limiting distribution.Simulation studies are presented to assess the performance of the proposedmethods, and the usefulness of generalized covariance models is illustrated byan analysis of the US stock market.

Covariance Model with General Linear Structure and Divergent Parameters