# Chapter 4: Structural Equation Models

### Identification of SEMs

Conditions for a (single-group) structural equation model to be identified are an extension of the corresponding conditions for factor analysis models. First, to identify the latent scales, it is sufficient to assume that the means and variances of all exogenous latent variables are set to 0 and 1 respectively, and all intercept terms (*α*'s) and residual variances (*ψ*'s) of structural regression models are also set at 0 and 1 respectively. Once this is assumed, remaining conditions for identification depend on the structure of the model. However, the following two-step condition is sufficient and usually easy to check:

- Re-express the model as a confirmatory factor analysis (CFA) model, by replacing all paths in the structural model (whether covariances or regressions) with covariances (two-headed arrows). Then use identification rules for CFA (see Chapter 2) to check whether this model is identified. If it is, the measurement model of the SEM is identified.
- Consider then the structural model on its own, and replace each latent variable in it with a single observed variable. If this model would be identified and if condition 1 holds, the whole SEM is identified.

A sufficient condition for requirement 2. to hold is that the structural model should be recursive. This means, roughly, that the model should not contain feedback loops. Such loops might take the form of regression models in both directions between two variables, a regression and a residual correlation between two variables, or longer chains of variables which imply similar loops which would allow us to start from a variable and return back to it by following the arrows in the path diagram. Non-recursive structural models are conceptually and practically complicated even when they are identified, so they should be avoided unless substantive theory gives very strong reasons to consider them.

[Bol89] gives a more detailed account of identification conditions for SEMs.

#### References

- [Bol89] Bollen, K. A. (1989). Structural equations with latent variables. Wiley.