"Bad" control variables in randomised treatment trial 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)?

As the other answer states, if you are interested in the effect of the treatment on money that does not work via value (the direct effect), then you would, in principle, like to control for value. If it Is the total effect (direct+indirect) you are after, you should not be controlling for variables that are (potentially) outcomes of the treatment. It is correct that you need to think about which effect you actually want to estimate.
However, value is a bad control and you will not be able to get an estimate of the direct effect that is as credible as one of the total effect. This is because there are probably other variables that cause both value and money (confounders) that you have not observed. Controlling for value would therefore introduce bias unless you are able to also control for these confounders.
It is usually very hard to make a credible case that you have controlled for all confounders. The beauty of randomised treatment assignment is that there are no confounders (for treatment), and you can therefore get a very credible estimate of the (total) causal effect of the treatment on the outcome. If you decide to try and estimate the direct effect, then you are back to worrying about confounding and the point of having run an experiment is kind of lost. Similarly, you would not want to throw away a good quasi-experiment by introducing bad controls.
Bad controls are less bad with a non-experimental research design. In that case you would have to worry about confounding to begin with and the “bad” controls would only mean additional potential confounders to take into account.
It is fine to use “demographic control variables” if they are determined before the treatment. However, you would more typically use these to demonstrate that randomisation has produced a treatment and control group that are balanced with respect to observed characteristics; e.g, a table with the means for both groups. If you have balance, the point of controlling for these variables would be to improve precision.
If you do, it should be in addition to your estimate without any control variables at all which is the main result of your experiment (unless you had pre-specified otherwise)
