We thank the reviewer for their detailed comments and helpful suggestions.

• There is no related work that is essential to understanding the (context for) key contributions of the paper but are not currently cited/discussed in the paper. However, there are some recent works [1,2] that also use high-order cumulants for identification of LiNGAM with latent variables. I recommend the authors discuss them.

We will include a discussion of these works in the related work section. Both of these papers focus on the problem of causal discovery—that is, learning the causal structure from observational data—rather than on the problem of causal effect identification. In the latter, the causal graph is assumed to be known, and the goal is to identify the causal effect of a treatment on an outcome (see, for example, the formal definition of this problem in the seminal work of Shpitser & Pearl, (2006).

In particular, [1] proposed a procedure for using high-order moments to test for the presence of an edge between two observed variables where there is exactly one latent confounder. It is noteworthy that Schkoda et al. (2024) extended this result to settings with an arbitrary number of latent variables.

In [2], the authors considered specific causal graphs under the assumption that for each latent variable $L$ in the graph, there exists a ``homologous surrogate" (see the definition in [2]). They then derived a set of structural properties using equations involving cumulants to recover the underlying causal structure. It is important to note that all the graphs considered in our work do not satisfy the assumption that every latent variable has a homologous surrogate except the graph $\mathcal{G}_1$, with a single latent variable and no edge from the proxy variable to the treatment.

Shpitser & Pearl, (2006) - Shpitser & Pearl, Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models, AAAI, 2006.

• It seems that the proposed method assumes that the underlying causal graph is known. Please correct me if I’m wrong.

As we mentioned above, in the problem of causal effect identification (the focus of the current work), the causal graph is known.

• Tramontano et al. (2024b) have already provided necessary and sufficient graphical conditions for generic identifiability, so this paper is not novel in terms of identifiability results; its novelty lies only in the identification method.

The identifiability results in Tramontano et al. (2024b) rely on the assumption that the full observed distribution is known. In contrast, our identifiability results require only knowledge of finitely many moments of the distributions, which is a strictly weaker assumption.

Moreover, the estimation method proposed by Tramontano et al. (2024b) is based on solving an overcomplete Independent Component Analysis (OICA) problem, which is inherently non-separable. This implies that the true mixing matrix cannot be recovered solely by optimizing for independence among the exogenous noise—precisely the approach taken by their algorithm. Consequently, their method fails to consistently estimate the correct solution. This limitation is evident in our experiments, where we observe that GRICA’s accuracy does not improve monotonically with sample size.

• (Minor) According to the experimental results shown in Figure 6, the proposed method is not superior to that proposed by Tramontano et al. (2024b) when the sample size is not very large.

One possible explanation is that cumulant-based methods process unbiased estimates of high-order cumulants (order 4 or higher), also known as k-statistics. While these estimators are unbiased, they tend to have high variance for small sample sizes.

In contrast, GRICA solves an optimization problem involving the $\ell_1$-norm of the observed samples, which have lower sample variance. As a result, for small sample sizes, GRICA may yield a lower mean squared error due to reduced variance. However, since the solution obtained by GRICA is not asymptotically unbiased, it cannot provide an asymptotically consistent estimator—unlike our proposed method.

We will add a remark on this point in the final version of the manuscript.

We thank the reviewer for their detailed comments and helpful suggestions.

• Lack of intuitive example or explanations for the main theorems. It could be better to move the proofs in the main text into the appendix and add more discussion.

We will use the additional page in the final version of the paper to provide further explanations and enhance the clarity of our theorems.

In short, all our theorems are based on the following intuition. Theorem 3.1 establishes that there exists a polynomial of degree $l+1$ (where $l$ is the number of latent variables in Fig. 1) whose coefficients can be computed using only the cumulants up to order $k(l)$. The roots of this polynomial correspond exactly to the causal effects of the variables $L_i$ for $1\leq i \leq l$ and $V_1$ on $V_2$. Solving this polynomial system reduces the space of possible solutions for the causal effect from $V_1$ to $V_2$ to a finite set of size $l+1$.

To further refine this solution and identify the correct causal effect uniquely, we need to derive additional equations that can only be satisfied by the true causal effect. This requires analyzing the nonlinear relationships among the cumulants of the observed distribution. Theorems 3.4--3.7 detail how this approach applies to the respective graphs.

• What is the key role of the additional variables (proxy and IV) compared with the results in [1]? How does it benefit the identification?

The problem addressed in [1] pertains to causal discovery—learning the causal structure from observational data—which is distinct from the problem of causal effect identification. Specifically, [1] seeks to recover all causal graphs consistent with the observed data, but multiple such graphs may exist, leading to different possible causal effects. In contrast, causal effect identification assumes that the causal structure is known and focuses on determining the effect of a treatment on an outcome. This distinction is fundamental in the literature (see, for example, the seminal work by Shpitser & Pearl, (2006). Following this framework, we assume that the causal graph is given and aim to uniquely identify the target causal effect.

Shpitser & Pearl, (2006) - Shpitser & Pearl, Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models, AAAI, 2006.

• Why does the method of 'Cumulant with Minimization' only occur in the experiments regarding the graph?

As mentioned in the final paragraph of Page 6, for the proxy setup shown in Figure 3 with a single latent variable (i.e., graph $\mathcal{G}_3$), we proposed an optimization-based technique that relies on computing lower-order cumulants compared to those used in the standard "Cumulant" method. We refer to this approach as ``Cumulant with Minimization", and it is tailored specifically to the structure of $\mathcal{G}_3$. As such, its results are presented only for graph $\mathcal{G}_3$ in Figure 6.

We thank the reviewer for their detailed comments and helpful suggestions.

• While the findings support the effectiveness of the proposed methods, applying them to a real-world dataset may further strengthen the validation and highlight their practical applicability.

Following the input of the reviewer, we have evaluated our methods on the data from Card and Krueger (1993), similar to the experiments conducted by Kivva et al. (2023). We provide a brief overview of the results here, while a more complete discussion will be included in the revised version of the manuscript.

The goal of the study is to estimate the effect of the minimum wage on the employment rate. In our experiments, we utilized the same preprocessing procedure in Kivva et al, (2023) and found that the results of our method, when assuming graphs $\mathcal{G}_1$ or $\mathcal{G}_2$, are consistent with findings in the existing literature. Specifically, the estimated causal effects under $\mathcal{G}_1$ and $\mathcal{G}_2$ are 2.68 and 2.71, respectively. Previous methods, such as the cross-moment approach (Kivva et al., 2023) and the Difference-in-Differences method, also provide an estimate of 2.68. In contrast, when assuming $\mathcal{G}_3$ as the true graph, we estimate a causal effect of 8.26. While this result still indicates a positive effect of the treatment on the outcome—consistent with prior work—the value of the estimated causal effect diverges from the ones reported in the literature. This suggests that, for this dataset, a causal graph excluding an edge from the proxy to the treatment (as in $\mathcal{G}_1$ or $\mathcal{G}_2$) may provide a better representation of causal relationships.

The code to reproduce the results can be found at https://anonymous.4open.science/r/CEId-from-Moments-20AC/estimation/real_data.ipynb.

Card and Krueger (1993) - D. Card and A. B. Krueger. Minimum wages and employment: A case study of the fast food industry in new jersey and pennsylvania, 1993.

Kivva et al., (2023) - Y. Kivva, S. Salehkaleybar, N. Kiyavash, A Cross-Moment Approach for Causal Effect Estimation, NeuriPS 2023.