When you look closely, there is no causal path from T to Growth, but there are two backdoor paths, both going through UC:
- one directly from T over UC to Growth
- and one from T over UC to dT to Growth
So, if indeed UC is unobserved in your study, there is no way to get an unbiased causal estimate for the effect of T on Growth from the data, given this DAG.
That said, adjusting for dT will close one of the two backdoor paths (the second one) and, since dT serves a proxy (or surrogate) variable for UC, it will also partially close the other backdoor path. (More precisely, only variation in UC that is unique to UC and not shared with dT will remain as confounding influence.)
More generally, this problem seems a bit ill-defined:
A DAG encodes your structural assumptions about the data-generating process. If you assume no (direct or indirect) causal effect of T on Growth, as is done in the DAG, the question of how to estimate that effect, given the DAG, becomes a bit nonsensical. It's zero by definition.
Regarding your question
For Multiple Linear Regression on the covariates, should all the covariates be taken into the account or only those in the adjustment set for the given exposure and outcome?
If you aim for a specific causal estimand and choose multiple regression as your estimation technique, then including the adjustment set in the model is the absolutely necessary minimum.
There can be benefits from including further variables which are not in the adjustment set (for example, you typically gain precision by including predictors of the outcome, even though they are not confounders) and there can be harms (for example, you might re-introduce bias by openining a previously closed path). This paper gives an excellent overview aobut these situations and the question what to adjust and what not to adjuts for.
Conversely, if I were to condition on C, the pathway Treatment -> C -> Outcome would be biased because C is on the front door path between the Treatment and the Outcome, so C should be left out from a regression model OR else B would also need to be conditioned on to close the formed back door path.
I don't think this is right. Controlling for B would not solve all the problems you would introduce by controlling for C. While it would help you close the back-door path Treatment -> C <- B -> Outcome, the front-door path Treatment -> C -> Outcome would still be closed.
what are the implications, benefits or drawbacks of including B-type variables in my regression models? Would I not gain any precision or explanatory power in the model by including it as a control, rather than optionally leaving it out?
Yes, that's right. Controlling for A allows you to identify the total causal effect of Treatment on Outcome unbiasedly. Additionally controlling for B should yield a more precise estimator.
Best Answer
Your graph is not a DAG because there is a cycle, so you're outside the model class.