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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.

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I'd advise you to read Bertanha's recent (not yet published) article that can be found at: http://economics.nd.edu/assets/153411/bertanha_marinho_jmp.pdf

Not only does it go over the standard normalization and pooling procedure, but it also provides a lengthy discussion on how to overcome the potential heterogeneity issues associated with this method.

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This paper, by Fort et al (2016) may help. Note though that stacking (same treatment, assigned at different cut-offs that depend on a sample's sub-groups - often the median within a village or a school) is different from pooling as discussed by Litschig & Morrison (different treatments, at different cut-offs). Yet, once the treatments (or treatment intensities) are the same, the two discussions are the same.

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