If you want to perform a covariance matrix you can use network analysis based on partial correlations.

Hello Loren,

Yes, you could do what you propose. However, the indices of model-data fit will almost certainly be different (since you're no longer evaluating whether/to what degree the control variables do or don't explain variation in your data set). The importance of those (control) variables to your specific research question(s) is likely the key consideration in making your choice.

Good luck with your work.

Hi, this is an interesting idea about which unfortunately I cannot tell much. How about making a small simulation in which you try that and compare it to a full model?

Equivalence, however, would assume that you'd have a full model in which all of the controls affect all other model variables which is not the best of the choices. Any restriction/fixation that you can defend makes the model more testable (albeit more challenging to fit) and, in turn, increases the causal support a bit. If you have instrumental variables in your set that you simply control, even increases the bias of model parameters:

Ding, P., VanderWeele, T., & Robins, J. (2017). Instrumental variables as bias amplifiers with general outcome and confounding. Biometrika, 104(2), 291-302. doi:10.1093/biomet/asx009

Hence, blindly throwing everything either into the model or in the controlling set is never a good idea and should be informed a the specification of a full causal model including the controls (on which the actually estimated SEM is based). This can be done with the open source software DAGitty with which you can play around with the controls, set up different plausible versions. The software will tell you (based on your rationale) which variables should be controlled, see e.g.,

Ferguson, K. D., McCann, M., Katikireddi, S. V., Thomson, H., Green, M. J., Smith, D. J., & Lewsey, J. D. (2020). Evidence synthesis for constructing directed acyclic graphs (ESC-DAGs): A novel and systematic method for building directed acyclic graphs. International Journal of Epidemiology, 49(1), 322-329.

Shrier, I., & Platt, R. W. (2008). Reducing bias through directed acyclic graphs. BMC Medical Research Methodology, 8(1), 70.

Vahratian, A., Siega-Riz, A. M., Savitz, D. A., & Zhang, J. (2005). Maternal pre-pregnancy overweight and obesity and the risk of cesarean delivery in nulliparous women. Annals of Epidemiology, 15(7), 467-474. doi:10.1016/j.annepidem.2005.02.005

HTH

Holger