reviews: 
Comments: **Post rebuttal update**: I'd really like to see a couple of real-world examples that might satisfy the assumptions, but my other concerns are largely addressed. I raise my score to 8 accordingly.

The paper considers the causal effect of an intervention $A=a^*$, where value $a^*$ is not observed in the data; A affects the outcome Y through unobserved variable Z, and covariates X relates to Z through an injective function g. The paper uses a conditional moment restriction (CMR) implemented by a kernel method called maximum moment restriction (MMR), and it then uses the control function (CF) approach to identify the outcome function between Z and Y. Both the CMR and CF depend on the assumption that A affects Z linearly. The approach is theoretically guaranteed, and experiments on synthetic datasets support the theoretical analysis.
Strengths: Using CF to achieve intervention extrapolation is a nice idea.

The theoretical analysis is serious and detailed (but I did not check the proofs in Appendix).

The paper is quite well written.
Weaknesses: *Technical novelties seem to be weak*. Theorem 4 seems to be an adaptation of the CF approach in (Newey et al., 1999), and Theorem 6 seems to be an adaptation of the IV approach in (D’Haultfoeuille, 2011). If there are some technical novelties, they should be discussed and compared to the original works; otherwise, I suggest being more explicit about this weakness.

*Some assumptions are strong*; particularly, the linear model between Z and A, and the injective model and noiseless model between Z and X. Intuitively, X is an observable proxy of the hidden Z, and the assumption means there is no information loss in this proxy, which is strong. Moreover,  both assumptions involve hidden variable Z and add difficulty to practical judgments. Since both assumptions are inherent in the current approach, I do not expect the author(s) to address this weakness in the rebuttal, but the following could certainly be done.

*Discussion of the setting and comparison to related work*. The discussion of the relationship to reinforcement learning is interesting but does not touch on when can we possibly expect linearity and noiseless injectivity. It would be more interesting to draw and discuss a couple of real-world problems that might satisfy the assumptions. 

On the other hand, (Khemakhem et al., 2020) and [1, 2], which are based on the former, are important related work that needs more discussions, and this would clarify the current approach.  For example, (Khemakhem et al., 2020) recover Z based on exactly the same graph as A→Z→X, and also assume g is injective but *allows an additive noise on X*; the identification also relies on assumptions on the A→Z part, where p(Z|A) is assumed to be an exponential family distribution but *allows nonlinearity*. Overall, I do not think the assumptions in (Khemakhem et al., 2020) are clearly stronger than those in the current work. Further, [1, 2] uses (Khemakhem et al., 2020) to estimate treatment effects, though not considering intervention extrapolation, [2] mentioned the ideas of CMR and CF in Sec 4.4.

[1] Wu, Pengzhou Abel, and Kenji Fukumizu. "$\beta $-Intact-VAE: Identifying and Estimating Causal Effects under Limited Overlap." International Conference on Learning Representations. 2022.
[2] Wu, Pengzhou and Kenji Fukumizu. Towards Principled Causal Effect Estimation by Deep Identifiable Models”. In: arXiv preprint arXiv:2109.15062 2021

*Additional experiments could be added*.

I think the ability to deal with unobserved confounders is a strength of the work. So, why not add experiments on this? I know eq24 contains hidden confounding, but this direction is not examined, e.g., by adjusting the strength of confounding and comparing to other methods.

As indicated above, adding real-world problems in experiments can greatly strengthen the work. Also, it would be interesting to replace the CMR part with iVAE (Khemakhem et al., 2020) and see how the results would change.

I will read the rebuttal and revised paper and raise my score to 8 if the issues/questions above are addressed.
Recommendation: 8: accept, good paper
Reviewer confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness/correctness: 3 good
Clarity: 3 good