Do we need Overlap/Common Support in case of a parametric regression? If I want to make a causal statement based on selection on observables. One typically assumes "Common Support" (/"Overlap") - which means that for any value of the confounding variables X a unit i can be potentially observed with treatment (D=1) and without treatment (D=0). 
I.e.: 
0<P(D=1|X=x)<1 

where P(D=1|X=x) denotes the probability of receiving treatment conditional dependend on the confounding variables X. In case of non-parametric (semi-parametric) estimation (matching on X or on the propensity score) this assumption is crucial. 
However, I am wondering whether this assumption has to hold if I want to estimate the treatment effect in a parametric regression (e.g. a simple multivariate linear model fitted by OLS). Then the assumptions on the error term (e.g. normality) enable us to extrapolate the results/treatment effects to regions without common support anyway - hence, we might also consider the units i without common support in the estimation(?). 
Please let me know about the necessity of this assumption in case of the parametric estimation. 
Thanks in advance.
 A: I am not sure if I agree with your characterization of extrapolation. It seems more about Xs than epsilons and normality.
The basic intuition of the support problem is that if you are going to estimate the counterfactual for a given person by someone matched to that person, then you need to find someone similar to that person in the counterfactual state. Matching makes it plain whether or not comparable untreated observations are available for each treated observation. In that it helps you to avoid identifying effects solely by projections into regions where there are no data points based on functional form assumptions. If your projections are good, this won't be an issue. If they are not, you've estimated something funny. For example, if the CIA holds, but linearity does not, then matching is consistent, but regression is not.
A related point is that matching weights observations differently than OLS in calculating the expected counterfactual for each treated observation. In OLS, all of the untreated units play a role in determining the expected counterfactual for any given treated unit. If you got some uncomparable ones, you're in trouble. In contrast, in matching only untreated units similar to each treated unit have positive weight in determining the expected counterfactual. This may mean that you have to throw away some data, but you get less bias in exchange for higher variance.
A: 
"Please let me know about the necessity of this assumption in case of the parametric estimation."

Traditionally, the concern has been model checking.  If you are going to rely on model extrapolation (or interpolation) then the least you can/should do is check that the functional form assumptions seem reasonable.  But you actually can't do the relevant checks if there are wide open spaces in the data where only treatment or only control cases live.
Obviously this applies as much to linear as it does to non-linear models, and also outside the context of causal inference.
