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.:
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.