SEM: saturated models versus just-identified models

Question

I read in Kline (2016) p. 147 that

A just-identified structural equation model is identified and has the same number of observations as free parameters.

I read here that

A saturated model is one in which there are as many estimated parameters as data points

Is it the case that saturated models are a more general class of models, which contain just-identified SEMs as a subset?

I understand from discussion in the linked SE thread that saturated models don’t necessarily have perfect fit, and understand from Kline’s book that just-identified models also don’t necessarily have perfect fit. That bit doesn't confuse me. However, I'm wondering:

Is a just-identified model simply a saturated model, but saturated model that also has the following features:

• It’s a structural equation model
• It’s identified

Is it the case that both saturated models and just-identified models by definition have zero degrees of freedom? The quotes above seem to indicate so, but I can't be completely sure the quotes are right or that I've understood them correctly.

Are there just-identified models that exist outside the context of SEM?

Kline, R. B. (2016). Principles and practice of structural equation modeling. Guilford publications.

Answer 1

Saturated models are just identified. I think just identified models are saturated (but it's possible that there is a just identified model I haven't thought of that is not saturated.) Both have zero df.

A regression model is a saturated / just identified structural equation model (as is anova, a t-test, etc).

You can think about these in very simple terms.

Let's say you have two variables, and all you want to estimate is their means: $y1$ and $y2$.

I estimate the means of these two variables. I have two parameters to estimate (the means), and two moments (data points), hence I have no df, and the model is saturated, or just identified.

If I add a constraint, for example:

y1 = y2

Or

y1 = 0

I've now overidentified the model, and it has one degree of freedom.

Let's say I try to add the mean of a third variable, $y3$ to my model. Now I have three unknowns, and only two data points. I can't estimate the model any more, it has -1 df, and is not identified.