In testing a structural model fit, how strongly do the measurement models fit influence the overall structural model fit? I am a beginner in SEM, sorry if this is an obvious question. But I have searched the literature and could not find anything that answers my question (except for this question: SEM: Do I still have to look at the fit of measurement models when overall fit is good? , but each latent variable only has 2 observed variables in that case and so it is clear that CFA would not make much sense).
My question is, should  a measurement model fit for each variable in a model (that is conducting CFA) be established first before moving on to the overall structural model?
Supposing that the CFA for the independent latent variable and dependent latent variable showed poor fit, does it still make sense to test the fit of the hypothesized structural model?
I am not sure whether this is a valid question. Because I could not find any study that reported initial measurement models fit before testing the structural model. 
 A: There are two questions here: (1) do you need to fit/examine a measurement model before fitting/examining a structural model? and (2) if a given measurement model exhibits poor fit, is it still worth fitting/examining a structural model. 


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*This is essentially a matter of could v. should. Could you just skip to fitting a structural model? Yes. Should you? Probably not. One reason: measurement models help to evaluate how reasonable your theory of construct measurement is. If you skip right to a structural model and get bad model fit, it could be because your theory of construct measurement sucks, or because your theory of structural construct relations sucks. In other cases, certain measurement models need to empirically tenable before you can make accurate inferences about structural relations. For example, if you want to examine certain structural parameters between groups, or over time, you would first need to establish measurement invariance between groups or over time, respectively (Little, 2013). You then compare your initial structural models against the more restricted of your measurement invariance models. Logistically, the difference in code between your final measurement model and initial structural model is likely so trivial, that I don't see any reason why not to look at a measurement model first.

*There is a method of accomplishing this goal--parcelling (see Little, Cunningham, Shahar, & Widaman, 2002, for an introduction)--though it is a somewhat controversial tactic, with vocal supporters/critics. Essentially, parceling entails grouping a set variables for a construct (let's say you have a 9-item questionnaire) into 3 aggregates (i.e., sums or averages). You then use these 3 aggregates as indicators of their respective latent variable, and repeat this process for each latent variable. With 3 indicators each, every latent variable in your model will be "just-identified", virtually guaranteeing a strong fitting measurement model. The rationale for parcelling is that it allows you to better evaluate the structural portion of your model, as essentially all model misfit will be on account of misspecified structural relations. The controversial aspect of it, is that some see it as a way of mathematically "tricking" your way into good model fit. Related to your first question, Little et al. (2002) therefore recommend first examining an un-parcelled measurement model, especially if there is not prior research strongly supporting your theorized measurement model. If, however, you're using a measurement model that has replicated consistently, you may be fine with using a parceled model right away. There are a number of other benefits/drawbacks to parcelling, and you really should just read the Little et al. (2002) paper if you're considering it--it's a pretty manageable read, even if you are new to SEM.  


References
Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.
Little, T. D., Cunningham, W. A., Shahar, G., & Widaman, K. F. (2002). To parcel or not to parcel: Exploring the question, weighing the merits. Structural Equation Modeling, 9, 151-173.
