When we have an RD with multiple cutoffs, and we pool all observations and estimate the treatment effect across cutoffs, what does the pooled estimate identify?
I have found one paper using a RDD with multiple cutoffs. Litschig and Morrison (2013) write: "Pooling requires the treatment intensity to be of comparable magnitude in order to interpret the size of estimated impacts. [Footnote: Treatment effects need not be the same across cutoffs. If treatment effects are heterogeneous, the pooled estimates identify an average treatment effect across cutoffs.]"
I'm looking for something more formal about what the pooled estimate of the treatment effect is. Even another paper with a longer or more formal discussion of the pooled estimate would be useful.
Reference: Litschig, Stephan, and Kevin M. Morrison. "The impact of intergovernmental transfers on education outcomes and poverty reduction." American Economic Journal: Applied Economics 5.4 (2013): 206-240.