Let's imagine I'm interested in studying the causal effect of beliefs in some ideas and behavior related to these ideas (say, if I believe sunscreen is good for my health, I use more sunscreen etc.). Then I run a survey asking several questions about beliefs and behavior for several items. Suppose I fit the following model:
y_ij = person_fixed_effect_i + item_fixed_effect_j + belief_ij + e_ij, e_ij ~ N(mu, sigma^2)
If I believe that person ideology and item ideology can be confounding, and assuming both person ideology and item ideology are fixed, how should I draw my DAG?
I think that both fixed effects capture every effect person and item ideology could have, and thus my DAG should be like this:
In this case, the model is identified and everything is good. This makes sense, since the fixed effect is supposed to capture everything non-observed and that does not change for individuals and items. However, I was wondering if this is correct, because the fixed effects capture many unobserved and constant variables. Because of this, perhaps it's possible for some of the effect of ideology to "escape" the fixed effect? So, my question is: Is my DAG correct? Or is it possible for person and items ideology to have a direct effect on belief, not mediated by the fixed effects?