Thank you for your detailed feedback and insightful questions. We appreciate the opportunity to address your concerns and provide clarifications. Below, we respond to the specific weaknesses and questions raised in your review:

Response to “Weaknesses”

1. Clarifications and improvements: Thank you for highlighting aspects of our work that needed clarification. To avoid redundancy, we refer to our responses to the questions below. Action: We incorporated all clarifications directly into the revised paper.
2. New discussion section: We appreciate your suggestion to expand our discussion and add a conclusion section. Action: We added a dedicated discussion section that includes a summary of key contributions, a discussion of limitations and their implications, and directions for future research.

Response to “Questions”

1. Implications of Eq. (3) and time-varying confounding: Thank you for this excellent question regarding bias due to time-varying confounding and methods like CRN.
   • Eq. (3) states that methods relying solely on the current history and future treatments to predict the $\tau$-step-ahead outcome will be biased for estimating CAPO/CATE. Intuitively, these methods ignore future covariates occurring after/during the treatment intervention, which act as "post-treatment confounders." If these covariates are omitted during training, they essentially become "unobserved confounders," introducing bias. Note that we are not the first to obtain this insight; this is a well-known fact in the causal inference literature [e.g., 1]. For unbiased estimation, proper time-varying adjustment mechanisms (e.g., iteratively integrating out post-treatment covariates as shown in Eq. (5) and Eq. (6)) are required.
   • Regarding CRN: The CRN can be interpreted as an instantiation of the PI-HA learner, using an encoder-decoder LSTM architecture. It learns a representation of the current history (encoder) and predicts $\tau$-step-ahead outcomes using this representation and the intervention sequence (decoder). However, because the CRN ignores post-treatment confounders, it is inherently biased and does not converge to the true CAPO/CATE, even with infinite data. Importantly, the PI-HA learner (and thus CRN) may still outperform other learners in practice due to trade-offs between asymptotic bias and finite-sample variance, which depend on factors such as sample size, time-varying confounding strength, and data complexity.
   • Regarding balancing: Balancing modifies the loss to learn a representation that is independent of the treatment (motivated by randomized trials). However, balancing does not address bias due to time-varying confounding as it ignores the post-treatment confounders. Instead, balancing aims to reduce finite sample variance (see [2]) and thus does not affect our asymptotic analysis (Theorem 1). Furthermore, balancing has the following drawbacks for the time-series setting: (i) guarantees currently only exist for the static setting [2]; (ii) balancing requires invertible representations, which is hard to fulfill in the time-series setting; and (iii) recent work has shown that balancing can even induce bias if representations are not invertible, so-called representation-induced confounding bias [3].
   • Action: We added a new Appendix F to discuss connections between time-varying confounding, asymptotic bias, finite-sample variance, and balancing.
2. Separate models vs. multitask framework:
   • In our work, we use separate models for each nuisance parameter. However, we acknowledge that a multitask framework could be applied, particularly for architectures like Transformers. Such a framework may perform better if the propensity scores and response functions share structural similarities (or worse if they don’t), as noted in [4, 5].
   • Our paper focuses on the theoretical and empirical properties of meta-learners rather than optimizing underlying nuisance models. Exploring multitask frameworks is an exciting direction for future work.
   • Action: We added a discussion on multitask frameworks and mentioned their potential benefits, which we added as a new research outlook to our revised paper.

Thank you for your constructive feedback and suggestions. We appreciate the opportunity to address your concerns and improve our paper. Below, we respond to the key points you raised:

Response to “Weaknesses”

1. Experiments:
   • Model-agnostic property and additional results:
     • We kindly point you to Appendix H, where we repeated experiments using a different model architecture (LSTM instead of transformer). We apologize that this was not visible enough in our main paper. These results empirically demonstrate that our meta-learners are flexible with respect to the choice of the underlying model. Action: In our revised paper, we provided a more visible reference to Appendix F.
     • Additionally, we would like to make the point that the "model-agnostic" property of our meta-learners does not even require empirical justification or proof. This property follows directly from the fact that our meta-learners are defined in terms of conditional expectations and probabilities. Estimating a conditional expectation, $\mathbb{Ε}[Y | X]$, is equivalent to minimizing the squared loss $\mathcal{L}(f) = \mathbb{E}[(Y - f(X))^2]$ over a sufficiently rich function class $f \in \mathcal{F}$. This equivalence is well-established in statistical theory and applies to any (machine learning) model class $\mathcal{F}$. Action: In our revised apper we elaborated on the term “model-agnostic”.
   • Choice of model for experiments:
     • Our experimental setup follows standard practices in the causal inference literature, where a fixed model architecture is used across all meta-learners. This approach ensures that performance differences can be attributed to the choice of meta-learners rather than the underlying models. Note that meta-learners have been studied already in a variety of other causal inference settings [e.g., 1, 2, 3, 4, 5, 6], and our paper follows a well-established way of setting up experiments in this community.
     • We deliberately chose a standard Transformer architecture for two reasons:
       1. A simpler architecture allows us to isolate and evaluate the properties of the meta-learners themselves.
       2. Introducing a more sophisticated architecture for one meta-learner (e.g., the causal transformer [7] as an instantiation of the PIHA learner) would necessitate equivalent architectures for all meta-learners to ensure a fair comparison. However, it is not fully clear what such architectures would look like e.g., for the DR-learner. We agree that optimizing model architectures for specific meta-learners is an interesting direction for future work but goes beyond the scope of this work.
     • In summary, our focus is on the theoretical underpinnings of meta-learners and providing practical insights (e.g., on what meta-learners to choose in which situations). Exploring more sophisticated model choices combined with our meta-learners is an interesting direction for future research but falls outside the scope of this paper.
   • Datasets:
     • We would like to clarify that we use multiple datasets for our evaluation, each with certain characteristics to show specific aspects of our theoretical results. While It is true that we only use one real-world dataset for our evaluation, we do this on purpose and follow established literature [e.g., 1, 2, 3, 4]. The reason is that proper evaluation of causal inference methods using real-world datasets is impossible due to missing causal ground truth. We thus follow established literature and report factual RMSE as a proxy for performance on real-world data, but focus on synthetic data for the main part of our experiments.
   • Action: In our revised paper, we now elaborate more clearly on the way we set up our experiments.
2. Improvements to later sections of the paper:
   • We appreciate your feedback on the later sections of the paper. Action: We have revised these according to your suggestions. Specifically:
     • Typos and grammatical errors: We have corrected the errors you identified.
     • Model Description: We followed your recommendation and moved parts of the detailed Transformer architecture description to the appendix.
     • Discussion and conclusion: We added a discussion section, which includes a summary of key findings, limitations, and an outlook on future work.
     • Experimental results: We expanded the discussion of our results, thereby adding more interpretations and linking them back to the theoretical analysis. For example, we now explain why the PI-HA learner performs well in certain scenarios despite being asymptotically biased (see also our response to question 3).

Thank you very much for your thoughtful review and positive evaluation of our paper! Below, we address your comments and suggestions in detail.

Response to “Weaknesses”

Based on your feedback, we have made the following improvements to the paper (see the changes in our paper in red color):

• We expanded Appendix E to include more details regarding nuisance parameters and the methods used to implement the corresponding models for their estimation.
• We added a new Appendix H, where we formally define the rates in Theorem 1 and provide explicit examples to make the results more accessible for a non-expert audience.

These additions aim to improve the clarity and accessibility of the paper for a broader readership.

Response to “Questions”

1. Precise definition of rates. Thank you for the suggestion. As outlined in our response to weaknesses, we added a new Appendix H where we formally define the rates in Theorem 1. This includes specific examples and additional context to make the assumption and its implications clearer.
2. Normalization in real-world experiments. Thank you for bringing up this important point. We used a standard normalization procedure, where we subtracted the mean $61.09$ and the standard deviation $14.48$. We would like to emphasize that our experiments aim to validate the theoretical results and compare the performance of different meta-learners relative to each other. As such, we do not claim absolute optimality of our results. There is indeed room for future work, such as combining our meta-learners with tailored state-of-the-art model architectures for specific datasets. Our paper lays the theoretical foundation for these future developments and provides practical guidelines for designing such algorithms.
   Action: We added details regarding the normalization to our paper. We also added a new discussion section, in which we outline possible future directions such as developing new models based on our meta-learners.

Thank you for your review and your helpful comments. As you can see below, we have carefully revised our paper along with your suggestions.

Response to “Weaknesses”

1. Motivation for meta-learners. We agree that the statement "meta-learners are state-of-the-art under the static setting" requires further elaboration. Causal inference is, at the fundamental level, a statistical estimation problem with nuisance parameters—HTEs cannot be estimated directly but depend on accurate estimation of response functions and propensity scores. The statistical theory underpinning such estimation problems builds on the efficient influence function (EIF) of the target estimand, resulting in model-agnostic estimators that are robust w.r.t. errors in the nuisance estimators [1, 2]. For finite-dimensional estimands (e.g., average treatment effects), these EIF-based estimators achieve efficiency bounds (i.e., are provably optimal) [1, 2]. For HTEs, recent work shows that they retain favorable properties as well (e.g., Neyman orthogonality and robust convergence rates) [3, 4, 5, 6, 7]. In the HTE literature, such model-agnostic learners are called meta-learners. Meta-learners are thus well-motivated by established statistical theory and have become standard in causal inference [e.g., 3, 4, 5, 6, 7]. For example, meta-learners are implemented in commonly used software packages for HTE estimation (e.g., EconML, DoubleML). In our paper, we close an important gap by extending and analyzing meta-learners for longitudinal data and time-series settings, and, thereby, we address challenges unique to these contexts.
   Action: We removed our statement "meta-learners are state-of-the-art under the static setting" and replaced it with a more detailed motivation.
2. Clarifying Figure 1. Thank you for pointing this out. We have added a concrete example to clarify why such edges might exist. For instance, consider heart rate or blood pressure measurements recorded over fine-grained time intervals. These measurements naturally exhibit temporal dependencies, with earlier states directly influencing later ones. We emphasize, however, that our methods do not require such edges to exist. In causal inference, assumptions correspond to the absence of edges, not (!) their presence.
   Action: We incorporated these clarifications into our problem setup.
3. Response surfaces vs. response functions. Thanks! You are correct, and we apologize for the ambiguity.
   Action: We have streamlined our terminology to consistently use "response functions" throughout the manuscript.
4. RA-pseudo-outcome (Eq. 8). You raise an important point. The time index of history information, $\bar{H}{t+1}$, ensures that the pseudo-outcome equals the CATE in expectation, i.e., ${\tau}^{\bar{a}, \bar{b}}(\bar{h}{t}) =\mathbb{E}[{Y}_\mathrm{RA}^{\bar{a}, \bar{b}} | \bar{H}{t} = \bar{h}{t}]$. We acknowledge that it is not immediately obvious why this is necessary.
   Action: To address this, we have added additional proofs in our new Appendix C.2, explicitly showing that all pseudo-outcomes considered in our paper are unbiased and can be used as second-stage regression targets.
5. Contributions. Thank you for pointing out the paper from Papadogeorgou et al. (2022). However, the paper focuses on a different target estimand—the average treatment effect of a stochastic policy (but not heterogeneous treatment effects). Furthermore, the proposed IPW learner is not our main contribution. Rather, we see our contributions in (i) the derivation of inverse-variance weights (Theorem 2) and (ii) the comprehensive theoretical analysis (Theorem 1), both of which are novel and specific to our setting.
   Action: We now cite Papadogeorgou et al. (2022) in our related work (Appendix B) and spell out clearly how our work is different and thus novel.
6. Propensities with high-dimensional history. You are correct that the propensity scores are defined with respect to the entire history of the respective time points, making them potentially high-dimensional. However, our meta-learners allow for arbitrary machine learning models to estimate the propensity scores, particularly those that handle high-dimensional data effectively. In our experiments, we used transformer-based architectures due to their known effectiveness in learning representations from high-dimensional time-series data. Specifically, we employed an encoder-only architecture with a causal mask to avoid lookahead bias. For further implementation details, we kindly point to Section 7.1 and Appendix E. Action: We expanded Appendix E and added additional details regarding the implementation of our nuisance functions. We are happy to answer any remaining questions you may have regarding our model choices.

7. (a) Model agnostic property. Please allow us to clarify the term "model-agnostic." You are absolutely correct that, in practice, performance is affected by the choice of model, and there may be an optimal model for specific data. This choice generally depends on factors such as the underlying data-generating process or sample size (e.g., for small samples, linear models might be preferred). By "model-agnostic," we mean that our meta-learners can be instantiated with arbitrary machine learning models, and our theoretical analysis does not depend on a specific model choice. For instance, our inverse-variance weights (Theorem 2) stabilize the DR loss regardless of whether a transformer, LSTM, or another model is used. Furthermore, the learners yield the same asymptotic rates (Theorem 1). Action: We clarified this in our revised paper.
   
   (b) Regarding our experiments: Our experiments aim to verify the theoretical results (e.g., Theorem 1) that characterize properties of our meta-learners, not of the underlying models. For this reason, we used the same transformer-based architecture for all our meta-learners because this ensures a fair comparison. Of note, this type of benchmarking is standard in causal inference [e.g., 4, 5, 6]. Furthermore, we would like to point to Appendix H, where we provided additional experiments with an LSTM instead of a transformer and observed consistent results.
8. Computational efficiency. Thanks for pointing this out. Let $R$ be the runtime of the underlying model (e.g., a linear model, transformer, etc…). The runtimes of the meta-learners are given in the following table and depend on how many models are required to be trained.
   Action: We added this to Appendix A. We also reported the exact runtime $R$ of our transformer-based implementation in Appendix E.

References

[1] Van der Vaart (2000). Asymptotic statistics. Cambridge University Press.

[2] Van der laan, Rubin (2007). Targeted maximum likelihood learning. International journal of biostatistics.

[3] Chernozhukov et al. (2018). Double/De-Biased Machine Learning for Treatment and Causal Parameters. Econometrics Journal.

[4] Foster, Syrgkanis (2023). Orthogonal Statistical Learning. The Annals of Statistics.

[5] Curth, van der Schaar (2021). Nonparametric estimation of heterogeneous treatment effects: From theory to learning algorithms. AISTATS.

[6] Kennedy (2023). Towards optimal doubly robust estimation of heterogeneous causal effects. Electronic Journal of Statistics.

[7] Nie, Wager (2021). Quasi-oracle estimation of heterogeneous treatment effects. Biometrika.