Testing for tau equivalence in a simple CFA model, based on correlation matrix

#1
For an assignment we are provided with only a (polychoric) correlation matrix. One of the questions concerns the different measurement models that can be tested (congeneric, tau equivalent, parallel).

My understanding of the course material would lead me to believe that analysing a correlation matrix, instead of covariance matrix would not allow you to differentiate tau equivalent measurement models form parallel ones.

IIRC, the diagonal of the observed variance--covariance matrix, i.e. the variances of the observed variables, is used to estimate the measurment error variances. In the case of a simple, 1 factor congeneric measurment model, the variance sigma_1 of indicator X_1 is parted into lambda^2, the variance attributable to the latent factor and theta_11, the residual variance. When analysing a correlation matrix, sigma_1 = 1 thus implies the relation theta_11 = 1 - lambda^2.

This means that constraining either the loading lambda or the measument error theta to be equal across indicators automatically constrains the other element to be equal too. In other words, we have to little information when using the correlation matrix to distinguish a tau equivalent measurment model (equal loading pattern, equal loadings) from a parallel measurment model (equal loading pattern, equal loadings, equal measurement errors).

However, when I try to fit a model to demonstrate that, I constrain the loadings to be equal, but the error variances do not become equal (see attachment View attachment 1453 for self-contained LISREL example).

Can someone point to the error of my reasoning/interpretation?