# Sharp Regression Discontinuity (RD) for one part of the sample, RCT for another?

I have a study where schools' 4th graders are randomized into treated and controls (at the school level, i.e. there are treatment and control schools). Within each treated school, treatment follows a sharp cut-off rule which also allows for a sharp regression discontinuity (RD) approach. In contrast, among 3rd graders, there are only treatment schools, but the RD approach can still be used: In any treated school, half of the students don't receive the program, strictly following assignment along a forcing variable.

I know how to set this up to exploit the RCT for 4th graders, the RD for 3rd graders, and to estimate everything in one go. I'm interested in doing that to increase power. However, this strategy mushes together the (4th grade) ATE and the (3rd grade) local RD estimate. Do you know of any literature that already discusses the properties of such a "combined" estimator, that uses an RCT for some part of the sample and an RD for another part? If not, what would be your arguments against such an approach?

• To clarify, do you mean that if the RCT was assigned 50/50, a 4th grader has a 25% probability of being in treatment (50% of being in treatment school and 50% of being assigned treatment within school) and a 3rd grader has a 50% probability (100% of being in treatment school and 50% of being assigned treatment)? – RickyB Nov 8 '17 at 15:18
• @RickyB To clarify: I was going to discard the ineligible 4th grade students in all schools, so eligible 4th graders would have a 50% chance of being assigned treatment (with treatment being assigned across schools). In contrast, I would keep all 3rd graders in treated schools only, and their treatment would be assigned within schools, at the eligibility cut-off. – user162760 Nov 9 '17 at 16:03