RDD estimates local average treatment effects around the cutoff point, where treatment and comparison units are most similar. The units to the left and right of the cutoff look more and more similar as they near the cutoff. Given that the design meets all assumptions and conditions outlined above, the units directly to the left and right of the cutoff point should be so similar that they lay the groundwork for a comparison as well as does randomized assignment of the treatment.
Because the RDD estimates the local average treatment effects around the cutoff point, or locally, the estimate does not necessarily apply to units with scores further away from the cutoff point. These units may not be as similar to each other as the eligible and ineligible units close to the cutoff. RDD’s inability to compute an average treatment effect for all program participants is both a strength and a limitation, depending on the question of interest. If the evaluation primarily seeks to answer whether the program should exist or not, then the RDD will not provide a sufficient answer: the average treatment effect for the entire eligible population would be the most relevant parameter in this case. However, if the policy question of interest is whether the program should be cut or expanded at the margin, then the RDD produces precisely the local estimate of interest to inform this important policy decision.
Note that the most recent advances in the RDD literature suggest that it is not very accurate to interpret a discontinuity design as a local experiment. To be considered as good as a local experiment for the units close enough to the cutoff point, one must use a very narrow bandwidth and drop the assignment variable (or a function of it) from the regression equation. For more details on this point see Cattaneo et al. (2018).