<p>No. There is not. I tried working on deriving one for a while. I struggled for a month, failed, and then emailed some contacts at the University of Chicago to ask if they had any ideas. My buddy asked Alex Torgovitsky who said something like &quot;the concept of a complier is quite tied to the existence of a binary treatment, so it's really not obvious how the result could generalize&quot;.</p>
<p>Good luck. I'd be delighted if someone has a better answer.</p>

CC BY-SA 4.0
2023-09-27T13:18:54.750Z
Thanks! Somewhat surprised about Torgovitsky's comment, since I thought for continuous instruments / treatments without controls there is a way to interpret the estimates as average causal responses. Angrist covers it in his IV slides p. 32 ff, there is also a section in MWE about it.
https://ocw.mit.edu/courses/14-387-applied-econometrics-mostly-harmless-big-data-fall-2014/02fdeb479fb5683f9d723db7b289a9d0_MIT14_387F14_Causaleffects.pdf Compliers are those moved from intensity s to intensity s'.
Papayapap
CC BY-SA 4.0
2023-09-27T15:36:46.860Z
Oh, you can get an interpretation as an Average causal response when estimating via 2SLS and a nonbinary treatment. Abadie's Kappa is about getting the characteristics of compliers though.
Michael Gmeiner
CC BY-SA 4.0
2023-09-27T16:52:31.207Z
Yea, that also. In my case, I am more interested to avoid the criticism that when including X linearly I will not get a positively weighted average of LATEs https://www.dropbox.com/s/7ub35oqau2if7qb/tsls-weights%20%281%29.pdf?dl=0. That can be also done with Abadie's Kappa and using IPW (in binary case).
Papayapap
CC BY-SA 4.0
2023-09-27T06:59:03.907Z
2023-09-27T06:59:03.907Z
Michael Gmeiner
<p>Does somebody know if there exist results and estimators to deal with controls in IV estimation under heterogeneous treatment effects, analogous to Abadie's Kappa (2003) for the binary case?</p>
<p>Abadie, Alberto, “Semiparametric Instrumental Variable Estimation of Treatment Response Mod-
els,” Journal of Econometrics, 2003, 113 (2), 231–263.</p>

<regression><instrumental-variable><causality><heterogeneity>
Abadie's Kappa (2003) for continuous treatments / IVs
Papayapap
# Abadie's Kappa (2003) for continuous treatments / IVs

(Asked by: username_0 on 2023-09-26)

Does somebody know if there exist results and estimators to deal with controls in IV estimation under heterogeneous treatment effects, analogous to Abadie's Kappa (2003) for the binary case?

Abadie, Alberto, “Semiparametric Instrumental Variable Estimation of Treatment Response Mod-
els,” Journal of Econometrics, 2003, 113 (2), 231–263.

**Tags:** regression, instrumental-variable, causality, heterogeneity

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## 1 answers

### Answer by username_1 on 2023-09-27. Score: 3

No. There is not. I tried working on deriving one for a while. I struggled for a month, failed, and then emailed some contacts at the University of Chicago to ask if they had any ideas. My buddy asked Alex Torgovitsky who said something like "the concept of a complier is quite tied to the existence of a binary treatment, so it's really not obvious how the result could generalize".

Good luck. I'd be delighted if someone has a better answer.

> **Comments:**
> **username_0**: Thanks! Somewhat surprised about Torgovitsky's comment, since I thought for continuous instruments / treatments without controls there is a way to interpret the estimates as average causal responses. Angrist covers it in his IV slides p. 32 ff, there is also a section in MWE about it. https://ocw.mit.edu/courses/14-387-applied-econometrics-mostly-harmless-big-data-fall-2014/02fdeb479fb5683f9d723db7b289a9d0_MIT14_387F14_Causaleffects.pdf Compliers are those moved from intensity s to intensity s'.  
> **username_1**: Oh, you can get an interpretation as an Average causal response when estimating via 2SLS and a nonbinary treatment. Abadie's Kappa is about getting the characteristics of compliers though.  
> **username_0**: Yea, that also. In my case, I am more interested to avoid the criticism that when including X linearly I will not get a positively weighted average of LATEs https://www.dropbox.com/s/7ub35oqau2if7qb/tsls-weights%20%281%29.pdf?dl=0. That can be also done with Abadie's Kappa and using IPW (in binary case).  

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