Kline (2016) writes on p. 135:
There are two kinds of path models. Recursive models are the most straightforward and have two basic features: their disturbances are uncorrelated, and all causal effects are strictly unidirectional. [...] Nonrecursive models have causal (feedback) loops or may have correlated disturbances.
On p. 137 in a section called "Implications of the distinction between Recursive and Nonrecursive Models", Kline continues
The assumptions of recursive models that all causal effects are unidirectional and that the disturbances are independent simplify the statistical demands for the analysis. For example, multiple regression can be used to estimate path coefficients and disturbance variances in recursive path models. The occurrence of a technical problem in the analysis is less likely for recursive models. It is also true that recursive structural models are identified, given that necessary requirements for identification are satisfied. [...] Another complication of nonrecursive models is identification. There are some straightforward ways to determine whether some, but not all, types of nonrecursive path models are identified. These procedures are described in the next chapter, but it is worthwhile to make this point now: Adding exogenous variables is one way to remedy an identification problem of a nonrecursive model. But this typically can be done before the data are collected. Thus it is critical to evaluate whether a nonrecursive path model is identified right after it is specified and before the study is conducted.
Often in books and talks have seen comments to the effect that the distinction between recursive and nonrecursive models is important or fundamental, but I always struggled to understand why. Am I right that in practical terms the key distinctions are as follows:
- Running nonrecursive models requires specialized SEM software, whereas recursive models could be done with multiple regression in more general statistical software (although in practice almost everyone would use SEM software for them anyway).
- "Technical problems" are more likely to happen with nonrecursive models. I'm unclear on what Kline means by "technical problems" - does this just mean that empirical underidentification is more likely to happen with nonrecursive models?
- Leaving aside empirical underidentification, a situation in which we have underidentification despite having non-negative degrees of freedom only happens with nonrecursive models, or is overwhelmingly more likely to happen with nonrecursive models than recursive models (if so, which of the two is it?).
Kline talks about the recursive/nonrecursive distinction being about path models, but from the definition it seems you could easily have a nonrecursive CFA just by having correlated disturbances. However, searching for "non-recursive CFA" reveals few hits. Is there a reason why this distinction is only or typically made in relation to path models?
Also, what's up with the naming of these types of models? How do the features of recursive and non-recursive models map onto the terms "recursive" and "non-recursive" respectively?
Kline, R. B. (2016). Principles and practice of structural equation modeling. Guilford publications.
