It's often stated that for analysis using an SEM technique, it is preferred to use an overidentified model compared to a just identified model. Why is that so ? My intuition says that for an over identified model will have more than one solution and after iteration if the converged solution fits the data well then only we can conclude something about the model as compared to the scenario with just identified model where we always get a perfect fit. Am I thinking about it correctly ?
The point of running an structural equation model is to be able to be wrong - and that's only true if it's over-identified (i.e. has degrees of freedom greater than zero). You can specify a multiple regression model as a structural equation model, you'll get the same answer, and the model will be just identified, so it will have zero degrees of freedom. But then what was the point of the structural equation model?
If a model is not identified, then it has more than one solution - it usually has an infinite number of solutions, all of which are equally good.
Here's an equation:
$x = 5$
There is one unknown, and one equation. It is just identified. That's fine if you want to know the value of $x$, but there is no way that the model can be wrong.
$x + y= 5$
There are two unknowns, and one equation. It's not identified. There are an infinite number of solutions and they are all equally good.
Here's a set of equations. There are two unknowns and two equations, it's just identified.
$x + y = 5$
$x - y = 1$
$x + y = 5$
$x - y = 1$
$2x + 2y = 6$
Now there are three equations, and two unknowns. It's now possible for this set of equations to be wrong, and it is wrong. But change that value of 6, and the model could be correct, and still be over identified.
Identification is a tricky issue, because the model must also be empirically identified, and that depends on the data.
Here we have two equations, and two unknowns, but the model is not identified.
$x * y = 0$
$x + y = 4$
There are two possible solutions, either x or y can be equal to zero.
So we want over-identified models, because they provide a single solution that can possibly be wrong. Just identified models also provide a single solution, but it cannot be wrong.
To expand beyond the question: Why is this a good thing? When I teach this, I say that you have a right to be sued. that's weird. Why would anyone want to be able to be sued? Surely you'd be happier if you couldn't be sued?
The ability to be sued means that people can trust you with things, because they can sue you if you screw up. You can rent a car, get a checkbook, sign a contract, get a credit card. Children can't do those things, because they can't be sued. In the same way, being able to be wrong has advantages - if you're right, it's better evidence for your theory.