Standard errors in covariance structure models: Asymptotics versus bootstrap

Commonly used formulae for standard error (SE) estimates in covariance structure analysis are derived under the assumption of a correctly specified model. In practice, a model is at best only an approximation to the real world. It is important to know whether the estimates of SEs as provided by standard software are consistent when a model is misspecified, and to understand why if not. Bootstrap procedures provide nonparametric estimates of SEs that automatically account for distribution violation. It is also necessary to know whether bootstrap estimates of SEs are consistent. This paper studies the relationship between the bootstrap estimates of SEs and those based on asymptotics. Examples are used to illustrate various versions of asymptotic variance-covariance matrices and their validity. Conditions for the consistency of the bootstrap estimates of SEs are identified and discussed. Numerical examples are provided to illustrate the relationship of different estimates of SEs and covariance matrices.