What Do I Do if My SEM Model Is Under-Identified?

If your model is under-identified, you have two primary solutions to fix this problem. First, you can reduce the number of proposed parameter estimates. This means that you can delete a covariance or structural relationship. Second, you can add more exogenous (independent) variables. By adding more exogenous variables, you increase the number of observations which could rectify your identification problem.

In My Opinion: Just-Identified or Near Identified Models

The closer you are to a just-identified model, the more skepticism reviewers have about your model unless you have a very simplistic model with few unobserved constructs. As stated earlier, a just-identified model will not allow you to assess how well your proposed model fits the data. Similarly, when you start approaching a just-identified model (df < 3), model fit indices start to bias upward.The bigger issue is a just-identified model or nearly just-identified model is often conceptualized where every construct is influencing every other construct in the proposed model. These types of models have unflatteringly been called “Big TOE” models, where the “TOE” represents “Theory of Everything”. A Big Toe model infers that everything is affecting everything else. Ultimately, that type of model is not discriminatory and just lets every construct have a relationship with every other construct in the model. These models are often met with skepticism, mostly because the theory underlining your proposed model is not going to justify that every construct in a model is interconnected and has a relationship. Having an over-identified model is a better option, as you can truly assess if the proposed model fits the data.

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.