# Interaction Term in Fuzzy RD

I'm hoping someone can help me understand the intuition behind the interaction term in a fuzzy RD model. The setup is as follows:

$x$ = rating variable with discontinuity at $x = k$

$D$ = dummy=1 if $x > k$

$T$ = treatment variable; the probability of treatment is higher for $x > k$

$y$ = outcome variable

Following the standard procedure of treating a fuzzy RDD as local IV model (see for example Fuzzy RDD issue, also implemented by the R/Stata rdrobust package), I would set up the following equations:

First stage: $T = \beta_0 + \beta_1D + \beta_2x + \beta_3D*x + \epsilon$

Second stage: $y = \beta_0 + \beta_1\hat{T} + \beta_2x + \beta_3D*x + \epsilon$

I'm confused as to why $D$ ends up in the second stage equation, given that $D$ served as our instrument and generally it would violate the exclusion restriction to have an instrument in the second stage. I understand that we generally need the interaction term in RD models to allow for a different relationship between x and y on either side of the cutoff, but I do not understand how we can "think of fuzzy RDD as a local IV model" when the "instrument" appears in the second stage.

Thanks for your time.

• There are relevant references in this post. May 11, 2018 at 2:07
• In a common effects world, RD looks very much like selection on observed variables where the selection variable, the running variable in this case, is observed by the econometrician. May 11, 2018 at 2:10
• Okay, so because this is RD, we already know that D only impacts y through its impact on x -- so it's okay to put it in the second stage? Is that a correct way of saying this? May 11, 2018 at 19:25