I am currently sitting on modeling for a paper and had actually assumed that I was well positioned with my model. However, in consultation with my supervisor, I now have doubts or doubts were sown. Since my supervisor has no idea at all about cross lagged models, I am looking for help here. My data consists of 3 waves in which I measure different constructs. Each of these constructs is used as a latent variable in my model. I have simplified my model here in the picture.
I have a set of questions, which form my independent latent construct X. I measured these, just like the mediator M and the dependent latent variable Y (set of questions), at 3 time points. I had previously read up on the topic of cross-lagged, however in such a model the relationship Y-->X would also be modeled, which is not supposed to be the case for me. Additionally, in a cross-lagged no effects within a wave are modeled.
My results are really very good for this model and fit my hypotheses. Nevertheless, my supervisor said that he has not seen the effects within a wave like this in a model and wonders if it can be done this way. The missing relation Y-->X is not a problem for him, but he still questions whether we are allowed to speak of a cross-lagged and whether the modeling can be done that way at all.
This would be my first question to the community. My second one goes in the direction that my supervisor said: If such a modeling is possible, then I would still have to use control variables such as age. Would this even be possible within the framework of this model. The interjection of a control variable is technically okay, but I don't know where I should insert it. The only thing that comes to mind is an additional mediator.

Thanks for your help in advance.
Christopher


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