I'm conducting a bi-factor confirmatory factor analysis using Mplus. The model modification indices suggest the model should correlate the residuals of latent variables. Somehow, I feel only observed variables have residuals not latent variables. Can latent variables have residuals?
@Erik Ruzek's answer is correct for the larger question - you shouldn't correlate your factors in a bifactor model.
For the smaller question - can latent variables have residuals, the answer is yes.
But don't think of them as residuals, think of them as unexplained variance. If two latent variables have a common cause, and that common cause is not included in the model, then some of the unexplained variance in each latent is shared - hence there should be a correlation.
If you are truly interested in a bifactor model, then you do not want to correlate the latent factors variances. It goes against the whole point of a bifactor model. As for your question, latent factors have their own variance and covariance(s) with other latent factors if you are running a CFA with more than 1 latent variables. MPlus is suggesting you free up the correlation between latent factors (not their residuals), as I understand it