Possibly unmeasured confounder and other highly correlated controls I am developing two models using observational data in which I have a binary outcome, a binary treatment and a series of confounders which I control for. The only difference is that the first model uses matched data (via exact matching), while the second uses the whole sample, with no matching.
Now, there's also a variable (to which I do not have access) that some colleagues point out might be a confounder. I am not convinced it is one, at least if we define confounders in line with Pearl and Cinelli's tradition (see Confounding variables in experimental study), namely "variables that affect both the treatment and the outcome". If we relax the definition and define confounders as those variables that correlate with treatment and outcome, instead, I acknowledge the unmeasured variable I was referring to can be considered a confounder.
I obviously know that - if this really is a confounder - my estimated effects turn out to be biased. Yet, among the other variables I control for, there are a couple that I expect would be highly correlated with this unmeasured variable. May I say that this somehow significantly reduces the issue of excluding this infamous unmeasured variable?
Also, to give a little more context, my outcome maps whether a person recovered from a given condition (this is just an example of my actual problem, so please do not focus on the theoretically relevant aspects of the health problem) and I know that the condition itself is highly clustered in space (aka neighborhoods) and the type of neighborhood is the unmeasured variable I was referring to.
Given that there is somehow almost perfect overlap between the "antecedent" of the outcome (the health condition) and the unmeasured variable (the neighborhood), and given that I have other variables that are highly correlated with the unmeasured neighborhood information, and assuming that a confounder implies correlation and not causation, do you think the estimates of the model can be considered reliable, while certainly pointing out the limitation? How would you proceed?
Thanks for your help!

Unpicking this, are you saying that:

*

*most people who suffer from the condition live in certain neighbourhoods

*the treatment helps most of those treated

*you do not have a record of which neighbourhoods your subjects live in

If so, then it seems me that the risk here is that the condition is actually mechanistically different in the minority who get it while not living in a high risk neighbourhood. When a doctor wants to treat, they will have information about the patient's address, and you will not be able to tell them the relevance of this.
The fact that you have recorded correlates of the neighbourhood does help, as you say, but does not abolish the concern.
Similar issues occur in clinical trials if they have a high exclusion rate, as for a recent Alzheimer's drug.
