Identifying informative correlation structure is important in improving estimation efficiency for longitudinal data. We approximate the empirical estimator of the correlation matrix by groups of known basis matrices which represent different correlation structures, and transform the correlation structure selection problem to a covariate selection problem. To address both the complexity and informativeness of the correlation matrix, we minimize an objective function which consists of two parts; the difference between the empirical information and a model approximation of the correlation matrix, and a penalty which penalizes models with too many basis matrices. The unique feature of the proposed estimation and selection of correlation structure is that it does not require the specification of the likelihood function, and therefore it is applicable for discrete longitudinal data. We carry out the proposed method through a groupwise penalty strategy which is able to identify more complex structures. The proposed method possesses the oracle property and selects the true correlation structure consistently. In addition, the estimator of the correlation parameters follows a normal distribution asymptotically. Simulation studies and a data example confirm that the proposed method works effectively in estimating and selecting the true structure in finite samples, and it enables improvement in estimation efficiency by selecting the true structures. The talk is based on joint work with Annie Qu.