Bad model fit for a second-order growth curve with time-varying covariates. Do I have to find a way out or will it depend on the aim of the analysis?

Question

I am running a second-order latent growth curve model with seven repeated measures. I have 2 time-invariant covariates (TIC, country and sex) and 2 time-varying covariates (TVC, both are latent variables (5 items each), seven repeated measures). All of the variables were measured at the same time.

I have already run an unconditional linear growth curve with my focal variable, let's say, A. The fit is good for this model. Later I ran a conditional model with the TICs. The fit decreased but was still good.

However, when I added the TVC, variables B and C, the model did not fit. (I did not add both at the same time. One by one as separate models) I understand that it means that both of the variables shifted their corresponding value of variable A in such a way that it does not fit a line anymore.

My first question is, if the aim of the analysis for the model with TVC is to see how much they shift the value of A in a certain direction, is it plausible to make an argument that "the model no longer fits a linear trajectory owing to the effects of the TVCs" and not respecify the model to make it fit? (or find a better-fitting model)

My second question is, does the worse fit for a linear model indicate that the effects of the TVC on variable A are inconsistent? I am assuming that if the effects of the TVC on variable A are constant, a linear trajectory should still fit.

I have a wald test code in my syntax to test this (I saw it in a textbook) but am not really sure how it works or if it is applicable here. Below is the syntax. (I use Mplus)

    AE1 ON AS1(P1);
    AE2 ON AS2(P2);
    AE3 ON AS3(P3);
 model test:
   P1=P2; P2=P3; 

Note. AE is my focal variable and AS is one of my TVC. The numbers following them indicate wave of data collection.

Thanks to everyone in advance who gave a thought to this post.

Answer 1

The misfit when you add additional variables with multiple indicators can have many reasons (especially in a complex model like yours), including problems with the measurement model (e.g., correlated error terms within or across constructs, cross-loadings). It is impossible to say what the reasons are without conducting a thorough analysis of your model, data, and results. Examining model residuals and/or modification indices can sometimes be helpful to locate the sources of misfit.