It has been pointed out in the literature that misspecification of the level-1 error covariance structure in latent growth modeling (LGM) has detrimental impacts on the inferences about growth parameters. Since correct covariance structure is difficult to specify by theory, the identification needs to rely on a specification search, which, however, is not systematically addressed in the literature. In this study, we first discuss characteristics of various covariance structures and their nested relations, based on which we then propose a systematic approach to facilitate identifying a plausible covariance structure. A test for stationarity of an error process and the sequential chi-square difference test are conducted in the approach. Preliminary simulation results indicate that the approach performs well when sample size is large enough. The approach is illustrated with empirical data. We recommend that the approach be used in LGM empirical studies to improve the quality of the specification of the error covariance structure.