I have been reading recently on fitting linear regression to evaluate causal effect of some treatment. Let's call the variable in the model representing treatment as Xj.
From what I have read, we need to make sure to include in the model other variables that affect both the responsible variable 'y' and the treatment variable Xj. I understand that only variables that affect both will impact the coefficient of Xj.
However, if a variable Xi impacts the response variable 'y' but is independent of Xj , isn't it still important to include it in the model since it can reduce the error ? It won't change the coefficient of Xj but it will affect its standard error which is important when trying to establish whether treatment effect is significant or not.
Is my logic incorrect ? Do we only need to worry about adding variables that affect both response and treatment variable ?