I can think of three reasons a model might contain correlated residuals. I'm sure there are many others. In CFA, correlated residuals among the factor indicators might represent a minor factor. In a growth model, auto-correlated residuals might be included. In an SEM model, a residual covariance between two factors might be needed because of an incomplete set of covariates in the regression of the two factors on a set of covariates.
Jon Elhai posted on Wednesday, January 27, 2010 - 11:23 am
This might be a relevant paper about correlated residuals: Cole, D. A., Ciesla, J. A., & Steiger, J. H. (2007). The insidious effects of failing to include design-driven correlated residuals in latent-variable covariance structure analysis. Psychological Methods, 12, 381-398.
If in a CFA the modindices indicate that a correlation of residuals (of two manifest scales which specify a latent construct) should be allowed in order to substantially improve model fit, this correlation certainly must be justified.
Is it indeed necessary to give a reason why those two (measurement) errors correlate? How could such a justification look like? Please give me an idea of typical explanations.