When fitting longitudinal CFA-based models (say one latent factor measured twice using the same indicators), it is customary to correlate the residual variances at time 1 with the residual variances from the same model at time 2. I often need a citation for this practice so here we go:
“Measurement residual variance contains variance due to measurement error as well as variance unique to an indicator but not common to the other indicators of the factor. Because measurement error is random, by definition, the measurement error component of the variance of the measurement residual cannot be correlated with other factors. The Specific variance component on the other hand… may have auto-correlation over time”
“if not taken into account, the stability of the model may be overestimated…… Although the inclusion of correlated measurement residuals.. could be decided on the basis of significance tests, they are generally included in longitudinal models a priori (Mheaton, Muthen, Alwin, & Summers, 1977). Except for the cost of the degrees of freedom, there is typically no harm in including these estimates.
- Newsom, J. T. (2015). Longitudinal structural equation modeling: A comprehensive introduction. Routledge. LINK