I am analysing the effect of a randomised treatment on several outcome variables.
First i am interested in whether the treatment changes the first outcome (non-pecuniary value) by controlling for demographics characteristics X. $$ Value = \beta_0 + \beta_1*Treatment_i + \beta_2*X_i + \epsilon_i $$
However, I am also interested if it affects another outcome (pecuniary value) that is according to the literature connected to the first variable $Value$. Therefor, I originally intended to control for it to avoid omitted variable bias.
$$ Money_i = \alpha_0 + \alpha_1*Treatment_i + \alpha_2*Value_i + \alpha_3*X_i + \epsilon_i$$
Now lastly, I am interested in the effect of the treatment on a third outcome variable (Probability to go to school) that is according to the literature related to the variables $Money_i$ and $Value_i$. Hence, therefor I thought to control for both of them in the regressions.
$$ Probschooling_i = \gamma_0 + \gamma_1*Treatment_i + \gamma_2*Value_i+\gamma_3*Money_i + \gamma_4*X_i + \epsilon_i$$
However, now I came across the concept of "bad" control variables that bias inference. In my case, as the variables $Value_i$ and $Money_i$ are themselves dependent variables in the other regressions, I thought that therefore they can be called bad control variables (some kind of dependencies). My question now is if that is true? Are they bad control variables so that i do need to take them out (and to not bias the treatment effect) ? Or can I just leave it as I have it (and so to avoid omitted variable bias)?