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    $\begingroup$ I think this answer and Jeremy's answer together do a good job of elucidating the major point of this question (+1). The model fit indices (and how they relate to sets of several simultaneous linear equations) were something I was considering discussing here in terms of their differences, but it seems you have covered that point already. Its interesting that SEMNET is currently exploding over this point (and has been for some time). $\endgroup$ Commented Dec 20, 2023 at 13:56
  • $\begingroup$ Could you give a simple example of an over-identified model, or a link to a discussion of one? E.g. if I add a node W and edges X->W, W->Z, would that be over-identified because the Y-W edge was missing? $\endgroup$ Commented Dec 20, 2023 at 18:02
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    $\begingroup$ Yes, or in your simple 3-variable model, the path model would be overidentified with 1 degree of freedom if you did not include/did not estimate the direct X --> Z path. Then, the model would have 1 degree of freedom for the test of whether the X --> Z regression slope (path) coefficient is equal to zero in the population. $\endgroup$ Commented Dec 20, 2023 at 18:22
  • $\begingroup$ If you have a saturated model but have binary variables in the mix, does the equivalence persist? I.e. does path analysis replicate logit regression (or probit)? $\endgroup$ Commented Dec 20, 2023 at 19:16