I'll write this as a thorough peer review. Given Florian is your co-founder, I'll aim for the kind of honest, constructive assessment a good referee would provide.

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# Referee Report: "Nonparametric Testability of Slutsky Symmetry"

**Journal:** [Top Field Journal in Econometrics, e.g., *Journal of Econometrics* / *Econometrica*]
**Recommendation:** **Major Revision**

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## Summary

This paper addresses a long-standing open question in nonparametric demand analysis: whether Slutsky symmetry — one of the two necessary and sufficient conditions for rationalizability of a demand system — can be tested without imposing parametric restrictions. The authors answer affirmatively, deriving a testable nonparametric restriction on observable conditional and marginal quantiles that is necessary for Slutsky symmetry in the IV demand model of Dette, Hoderlein, and Neumeyer (2016). The key theoretical engine is a bivariate extension of the Hoderlein-Mammen (2007) identification result for average partial effects in nonseparable models, which the authors show acquires non-trivial correction terms absent in the univariate case. They also connect their result to nonparametric welfare analysis with multiple goods, where these correction terms govern the gap between observable quantile demands and true Hicksian demands.

The question is important, the result is non-trivial, and the refutation of the Maes-Malhotra (2024) conjecture is a genuine contribution. I recommend major revision on the grounds detailed below.

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## Assessment of Contribution

The paper's core contribution is credible and significant. The testability question for Slutsky symmetry has remained open despite active work on the negative semidefiniteness side (Dette et al., 2016) and despite the recent pessimism generated by Kono (2025) and Maes-Malhotra (2024). The authors correctly identify that the difficulty is not impossibility but rather the need to move from univariate marginal quantiles to a combination of conditional and marginal quantiles — a subtle but consequential shift.

The bivariate extension of Hoderlein-Mammen (2007) in Lemma 2.1 is technically interesting in its own right. The appearance of the Dij terms — which capture how the composition of preference types at a fixed demand level shifts with prices — is a genuinely new economic insight, not merely a technical artifact. The connection to welfare analysis in Section 3, while brief, correctly flags that existing two-good results (Hausman-Newey, 2016) do not extend cleanly to multiple goods and that the correction terms introduced here are the precise objects governing that gap.

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## Major Concerns

**1. Necessary but not sufficient: the gap between the condition and a test**

The paper's title and abstract claim to establish "nonparametric testability" of Slutsky symmetry, and the condition in Theorem 2.1 is explicitly described as *necessary* for symmetry. However, the paper stops short of establishing *sufficiency* or providing an explicit testing procedure. A testable necessary condition alone does not establish testability in the sense typically understood in the literature (i.e., one can reject rationality with positive probability when it fails, and cannot reject when it holds). The paper should either:

(a) Clarify and defend the sense in which the condition is "testable" — for instance, by characterizing the power of a test based on (2) against local alternatives — or

(b) Explicitly acknowledge that the paper establishes a *necessary condition* and that construction of a fully-powered, asymptotically valid test is left to future work, and adjust the title and claims accordingly.

This is not a minor terminological issue. The abstract states "we derive nonparametric conditional quantile restrictions... that constitute a testable implication of Slutsky symmetry." Whether an implication is testable depends not only on its form but on whether consistent estimation of its components and inference on their equality is feasible. The paper gestures at this in the conclusion ("all of its components are estimable from data via conditional quantile inference methods") but does not verify it. Estimation of high-dimensional conditional quantile functions, their price derivatives, and the density-ratio correction terms (Cij, Dij) is nontrivial, and the curse of dimensionality is severe for L > 3.

**2. The correction terms: identification, estimation, and magnitude**

The correction terms Cij and Dij are central to the main theorem and the welfare discussion, yet their properties receive limited treatment.

- *Identification:* The Dij terms involve gradients of the conditional distribution FYi|W,Yj=y*j with respect to prices, holding the conditioning value fixed. These are not standard objects in nonparametric econometrics. The paper should clarify how these terms are identified — specifically, whether Assumption 2.2 and the regularity conditions in A.1 are jointly sufficient for point identification of Dij(w*) from observable data, or whether additional restrictions are needed.

- *Estimation:* Even granting identification, consistent estimation of ∂p FYi|W,Yj=y(·) requires nonparametric conditional density estimation with endogeneity correction (via the control function V). The rates of convergence will be slow. A brief discussion of the estimation strategy and feasibility — even at a heuristic level — would substantially strengthen the paper's claim that the condition is empirically usable.

- *Magnitude and economic relevance:* Section 3 provides useful economic intuition for when Cij ≈ 0 (weak separability) and Dij ≈ 0 (approximate monotonicity in preferences). But without simulation evidence or an empirical illustration, it is hard to assess whether the correction terms are quantitatively important in realistic demand systems or whether a simplified version of condition (2) that sets them to zero would have adequate power. Even a stylized parametric example — say, a translog demand system with known heterogeneity — would illuminate this.

**3. Relationship to Maes-Malhotra (2024) conjecture**

The paper describes the Maes-Malhotra conjecture as claiming that "nonparametric testability of Slutsky symmetry is impossible" in the Dette et al. setting, and presents Theorem 2.1 as a refutation. However, the paper does not provide a detailed analysis of *why* the conjecture is wrong — that is, what aspect of their argument fails in light of the present results. This is not merely scholarly courtesy: understanding the source of the disagreement would help readers assess the robustness of the present approach and whether the conjecture might hold under slightly different assumptions (e.g., with less regularity). A paragraph or appendix section carefully diagnosing the gap between the two papers is warranted.

**4. The d-dimensional extension**

Lemma 2.1 is stated for the bivariate case (L ≥ 3), with the comment that extension to d dimensions follows by repeated conditioning but "becomes even more complicated." Since the welfare analysis in Section 3 applies to general L-good systems, it would be useful to at least state the structure of the d-dimensional condition formally, even if proofs are omitted. The claim that the test applies to multi-good demand systems (L > 3) rests on this extension, and readers cannot assess its plausibility from the current paper.

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## Minor Concerns

**5. Assumption 2.2 and endogeneity**

The independence condition A ⊥⊥ (P, X) | Z is carried over from Dette et al. (2016) but deserves brief re-motivation in the present context. In particular, the appearance of Dij is driven precisely by the fact that conditioning on Yj = y*j induces dependence between Yi and (P, X) even under Assumption 2.2. A sentence explicitly flagging this tension — and confirming that the assumption is nonetheless sufficient for the identification result — would prevent confusion.

**6. The role of the instrument S**

The paper notes that assuming a single instrument S is "for ease of exposition" and can be generalized (citing Gunsilius, 2023). This is fine, but the paper should at minimum note how the dimension of S interacts with identification of the Dij terms. If Dij requires additional exclusion restrictions beyond those needed for identification of marginal demands, this is relevant for applied work.

**7. Theorem 2.1 is a necessary condition only**

The theorem statement should make clearer — perhaps by adding a "Remark" immediately following — whether the condition in (2) is also sufficient for Slutsky symmetry, or whether there exist demand systems that satisfy (2) but fail symmetry. If it is only necessary, the gap between the restriction and the symmetry condition itself should be characterized, even informally.

**8. Presentation of correction terms**

The correction terms in Theorem 2.1 are notationally heavy, which is unavoidable given the problem's structure. However, the paper would benefit from a schematic summary — perhaps a displayed equation showing the structure of (2) before unpacking the correction terms — that helps readers parse the main result before being confronted with the full notation. A worked example with L = 3 (two goods plus numeraire) would also substantially improve accessibility.

**9. Proof of Theorem 2.1**

The proof invokes the disintegration theorem (Chang-Pollard, 1997) to justify the uniqueness of the two sequential conditioning decompositions at a point (y*i, y*j). This step is correct but somewhat abruptly stated. The paper should briefly verify that the conditions of Chang-Pollard Theorem 1 are satisfied under Assumption A.1 — specifically, that the relevant measure is Radon on R^(L-1). The current footnote-level treatment ("the conditions of Theorem 1 in Chang and Pollard (1997) are satisfied since our measure is Radon on R") is insufficient for a result of this centrality.

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## Recommendation

The paper addresses a genuine open question and contains real theoretical contributions. However, the gap between establishing a necessary condition and claiming testability requires bridging — either by developing an explicit test with asymptotic guarantees or by substantially tempering the claims. The paper also needs more careful treatment of the estimation feasibility of the correction terms and a clearer engagement with the Maes-Malhotra conjecture. I encourage the authors to revise and resubmit.

**Decision: Major Revision**

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*Additional note for the authors:* The welfare analysis in Section 3 is the most immediately applicable contribution for empirical researchers, and it is currently the shortest section. Expanding it — with a concrete characterization of the correction factors in a tractable demand system — would significantly broaden the paper's appeal beyond the theory audience.