I'm uncertain exactly what the effect of fixing factor loadings across levels is. My understanding that if factor loadings are constrained to be equal across within and between level for endogenous latent variables, this indicates the latent variable has a random intercept, as do the observed indicators of that factor. Is this correct? And if the factor loadings are not so constrained, this indicates that they have random intercepts but the latent variable does not, yes?
Also, if one wishes to fix the variances of latent variables to 0, is it correct to fix the factor loadings to be equal across levels? Or should they be fixed to be proportional, using a constant? I've read a bit about this on the board but I'm not sure I'm interpreting it correctly.