{"id":"8984f8ce-8881-4f50-8141-010781c3f52b","arxiv_id":"2608.00136","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new nonparametric test detects whether people are influenced by their peers' traits or outcomes, catching nonlinear spillovers that linear regressions miss.","lead":"This paper proposes a nonparametric test that detects whether people or firms influence each other through their characteristics or outcomes, even when the influence is nonlinear. The test only needs the usual linear regression under \"no influence\" and could change how economists check for peer effects.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Outcome-spillover rejections rest on an unstated instrument set; without a valid Z, Theorem 1's null distribution need not apply.","rationale":"The paper's central theoretical claim is that S is asymptotically standard normal under the approximate null and consistent against nonparametric alternatives, with estimation only under the null. The proof structure is plausible: the leading term of the gradient is a linear form in Z'ε, and the remainder vanishes at the stated rate p^3/n → 0. I found no internal contradiction in the rate conditions or the CLT argument. The most load-bearing assumption for the outcome-spillover claims is therefore validity of the instrument matrix Z, because the y and cy tests are built from functions of the endogenous peer outcome w_i'y. Assumption 3 requires only uncorrelatedness of ε with Z; this is enough for the linearized leading term, but only if Z is genuinely exogenous and does not itself contain ψ(w_i'y) via the OLS special case. In the four empirical demonstrations, the instrument set is never reported for the y/cy rows, so a reader cannot tell whether the asymptotic null distribution applies. This is precisely the concern identified by the reader, and it is concrete and fixable: the authors can disclose Z, or rerun the applications with an explicit excluded-instrument strategy. The recommended s-test union rule also lacks size control, but that is secondary because the individual test results are already conditional on Z validity. I therefore agree with the reader's conditional verdict: the theoretical core is credible, but the empirical support for outcome spillovers is not decisive until the instruments are specified and validated.","tokens_in":32894,"tokens_out":10812,"duration_ms":127192,"concrete_test":"For each row of Tables 1–4 that uses y or cy, obtain the exact n×m matrix Z (explicit list of columns) and the estimation routine. If Z=U, recompute the test using an external instrument set—for randomized designs, predetermined peer attributes and their cross-products; for network designs, Bramoullé-style peers-of-peers covariates—and check whether the same rejections survive. Additionally, run a placebo simulation under the null y_i = x_i'β + ε_i with the reported W and the reported Z: empirical rejection rates for c, y, and cy should be near 5%, and the s-test union rule should not materially exceed the nominal level.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The y and cy tests include basis functions ψ(w_i'y) in the matrix U. Under the null, y_i = x_i'β + ε_i, so w_i'y = w_i'Xβ + w_i'ε. For S to be asymptotically standard normal, the proof requires Z'ε/n to be centered and to satisfy a CLT; Assumption 3 only states that ε_i and z_j are uncorrelated. If Z is taken to be U — the OLS special case explicitly allowed after equation (2.7) — then Z includes ψ(w_i'y), which is correlated with ε through the nonlinearity, so E(Z'ε) ≠ 0 and Theorem 1 does not apply. If instead Z is an external instrument matrix, the applications must state its construction. However, Tables 1–4 report many rejections through the y/cy channels but never list the instrument set Z for those tests. Without knowing whether Z is valid, the empirical rejections could be driven by endogeneity (e.g., common unobservables or simultaneity) rather than by spillovers. This is the load-bearing gap: the outcome-channel support is unverifiable as reported. A secondary compounding issue is that the s-test union rule in Definition 1 has no calibrated size, which can inflate rejection rates even under the null.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a nonparametric LM-type specification test for cross-unit spillovers. Spillovers may enter through peers' attributes (c), peers' outcomes (y), or both (cy), represented by unknown functions approximated by a divergent series expansion. Under the approximate null of no spillovers, the test statistic S in (2.11) is claimed to be asymptotically standard normal, and the test is claimed to be consistent. Estimation is required only under the null, which is an attractive feature. The paper also defines a composite 's-test' rejection rule based on the c, y, and cy tests, gives practical guidance on choosing the number of basis functions, and illustrates the method on four empirical applications.","tokens_in":33269,"tokens_out":5227,"duration_ms":58078,"significance":"If the central claims hold, the paper provides a practically valuable diagnostic: a researcher can test for potentially nonlinear spillovers while estimating only a linear regression under the null, with cluster-robust standard errors. The series/LM construction is standard, and the paper is careful to provide cluster-robust variance formulas, Monte Carlo evidence, and extensions to heterogeneous and noisy interaction structures. The empirical applications are relevant and show that the test can sometimes detect dependence missed by linear specifications. These strengths are undercut by several load-bearing issues: the recommended tuning parameter violates the rate condition required by the theorems; the composite s-test has no stated or simulated overall size; the outcome-channel tests require an instrument set that is never described in the empirical work; and the core null CLT is imported from self-cited work without statement. These issues are fixable within the scope of the manuscript, so I regard the contribution as promising but not yet fully supported.","major_comments":[{"comment":"The recommended tuning choices are p_c = [n^{1/3}/l], p_y = [n^{1/3}], and p_cy = [n^{1/3}/(l+1)]. For fixed l, each is Theta(n^{1/3}), so p^3/n = Theta(1), not o(1). Theorems 1 and 2 and the proof of Theorem C2 explicitly require p^3/n = o(1). Thus the implementation recommended in Section 4 is outside the range covered by the asymptotic theory. The statement that p^3(l+1)/n -> 0 is 'equivalent' to p^3/n -> 0 because l is fixed is incorrect at the recommended rate.","section":"Section 4, Eqs. (4.1)-(4.3); Theorem 1"},{"comment":"The s-test rejection rule rejects when at least one of the c, y, or cy tests rejects. No size analysis is supplied for this union rule. Even if each component test has asymptotic size alpha, the overall null rejection probability is not alpha and may be substantially larger, especially since the three tests are correlated. The Monte Carlo study in Appendix A reports size only for the cytest; there is no simulation of the s-test's overall size. Because all four empirical applications report 'stest result', the reported rejection frequencies have no stated null calibration.","section":"Definition 1 and Section 4"},{"comment":"For the y and cy tests, valid instruments Z are required so that E(Z'epsilon)=0. The paper says OLS is a special case with Z=U, but when U contains psi_i(w_i'y), this fails under the null: y_i = x_i'beta + epsilon_i implies w_i'y depends on w_i'epsilon, so Z'epsilon is not mean-zero and Assumption 3's uncorrelatedness condition does not hold. The four applications report many rejections through the y and cy channels but never state what instruments Z are used for those tests. Without that information, the outcome-channel rejections in Tables 1-4 cannot be verified and could be driven by endogeneity rather than spillovers.","section":"Section 2, after Eq. (2.7); Tables 1-4"},{"comment":"The final step of the proof of Theorem 1 invokes Theorem A.1 of Gupta, Qu, Srisuma, and Zhang (2025) without stating it, and Lemma D1 is imported from Gupta et al. (2025) without proof. That CLT for growing-dimension quadratic forms is the main engine of the standard-normal result. Because the cited theorems are from the authors' own prior work and are not reproduced, the derivation is not self-contained and the conditions are not transparent to the reader. State the required result or provide a proof.","section":"Proof of Theorem 1, Section C.2"}],"minor_comments":[{"comment":"The notation 'p = max{2, [n^{1/3}]}' is slightly confusing because the lower bound already ensures p>1, and the sentence 'always want p>1' is redundant with the max.","section":"Section 4, Eq. (4.1)"},{"comment":"There is a duplicated sentence fragment: 'components in the same cluster. components in the same cluster.'","section":"Appendix A, footnote 11"},{"comment":"The note says standard errors are clustered at the department level, while the reproduced original specification in Appendix E.4 says clustering is at the university-year level. Clarify which clustering is used in the stest.","section":"Section 5.4, Table 4 note"},{"comment":"The explanation of how w_i^{race} is obtained from W^{race} is terse and would benefit from a more explicit description, especially since the full-sample specification is not an affine function of the peer-race variable used elsewhere.","section":"Appendix E.2, footnote 15"},{"comment":"The paper would be easier to follow if the instrument matrix Z were explicitly defined in one place for the three tests, including the exact number of instruments m and how the instruments differ between the c test and the y/cy tests.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The core idea is sound and the paper has clear practical value, but the recommended tuning rate directly contradicts the main theorem, the s-test has no size calibration, and the outcome-channel empirical evidence is missing the instrument set. All of these are fixable without changing the main approach. I would not reject, but I would require a substantive revision addressing these issues before a further assessment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper builds a usable LM/series test for zero nonparametric spillovers through peers' attributes and/or outcomes, and it only requires estimating the null linear model. The c-channel part is probably fine. The y-channel part, as written, has a gap that threatens the empirical claims: the applications never state what instruments Z are used, and the paper explicitly allows OLS (Z=U), which cannot satisfy the orthogonality condition when U contains basis functions of other units' outcomes.\n\nWhat's genuinely new: adapting Hong-White/RESET series diagnostics to network spillovers, cluster-robust asymptotics, and a noisy-network extension. The proof structure is standard, and the simulation study plus four applications are real work. The test could be genuinely useful for applied peer-effects work.\n\nMain soft spots: (1) The recommended p ≈ n^{1/3} contradicts the theorem's p^3/n → 0; with p = n^{1/3}, p^3/n → 1, so the asymptotic normal approximation isn't justified. This is an internal inconsistency, and the paper should either give a rate-compatible rule (e.g. p = c n^{1/3 - δ}) or prove the result under larger p. (2) The outcome-spillover tests require instruments for w_i'y; Assumption 3 only states ε_i ⟂ z_j. If Z=U, then z includes ψ(w_i'y) = ψ(w_i'Xβ + w_i'ε), which is correlated with ε, so E(Z'ε) ≠ 0 and Theorem 1 doesn't apply. The empirical tables report many y/cy rejections but never list the instrument set. Without that, those rejections could be driven by endogeneity rather than spillovers. The c-channel results are not affected by this. (3) The union rule for the s-test has no calibrated overall size; rejecting when any of c, y, cy rejects will typically inflate type I error, and the simulations only report the cy test. This is fixable, but currently the recommended decision rule lacks a size guarantee.\n\nMinor: Theorem 1 imports a CLT from the authors' own related paper; that's acceptable, but it makes the paper not fully self-contained.\n\nBottom line: a useful practical diagnostic, and the c-channel logic holds up. But the y-channel as reported is not verifiable, and the tuning recommendation contradicts the theorem. The paper deserves a serious referee; I'd push for major revision—clarify or restrict the OLS case, report instruments in every empirical table, fix the p-choice, and calibrate the union rule.","headline":"A useful-looking c-channel spillover test with an unverified outcome-channel instrument story and a p-choice that contradicts its own theorem.","tokens_in":33679,"tokens_out":3090,"would_cite":false,"duration_ms":36062,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62G20","62P20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A simple LM-style statistic, computed under the null of no spillovers, detects nonlinear cross-unit spillovers and is asymptotically standard normal.","keywords":["Nonparametric test","Cross-unit spillovers","Social interactions","Interference","Peer effects","Series approximation","Cluster-robust inference","Specification test"],"falsifier":"Run two simulations: (1) under the null of no spillovers but with an instrument that is correlated with a cluster-level shock, check whether the rejection rate of S stays near 5%; (2) under a known nonlinear spillover f(s)=s² with valid instruments, check whether power rises toward 1 with n. The paper reports only a size simulation with valid instruments, so neither of these checks is documented.","tokens_in":32829,"feed_emoji":"👥","tokens_out":6720,"duration_ms":70249,"temperature":0.7,"pith_summary":"The paper aims to establish that cross-unit spillovers of unknown nonlinear form — through peers' attributes, peers' outcomes, or both — can be detected by a single test statistic S. The test is an LM-type diagnostic built from the gradient of an IV objective under the null of no spillovers, so a researcher only needs to estimate a standard linear regression. The authors prove that S is asymptotically standard normal under the null and consistent against nonparametric alternatives, under cluster-robust variance and a growing number of series terms. If correct, applied researchers can screen for nonlinear interference without committing to a parametric spillover model. Four empirical replications show the test frequently rejects where linear peer-effect regressions do not.","feed_headline":"Nonlinear spillovers caught by a test built from a linear regression","feed_subtitle":"Standard-normal statistic for peers' attributes or outcomes; it rejects in settings where linear models find nothing.","key_machinery":"The central object is the series-expanded gradient vector d̂ = −(2/n) U' P_Z (y − X β̂), where U stacks basis functions ψ_i(w'y) and ψ_i(w'c_j) (Hermite polynomials in the applications) and Z is an instrument matrix. The statistic S is a cluster-robust quadratic form in this gradient, centered and scaled by q = p(l+1) restrictions. Because the gradient is evaluated under the null, no nonparametric estimation of the spillover functions f or g_j is needed; the basis functions act as a growing-dimensional sieve that makes any departure from the null detectable in the projected moment conditions.","core_discovery":"Under the approximate null H0A of no spillovers (all series coefficients µ equal zero), the statistic S = (n d̂' Ĥ^{-1} d̂ − q)/√(2q), built from the gradient of a series-approximated IV objective evaluated at the restricted linear estimate, converges in distribution to N(0,1) as n and p grow with p^3/n → 0, and S is consistent against alternatives where any series coefficient is nonzero. The test simultaneously captures attribute-channel (c), outcome-channel (y), and joint (cy) spillovers, and requires only estimation of the null model y_i = x_i'β + ε_i.","pith_inferences":["Not stated in the paper: a rejection by S only indicates that some nonlinear dependence exists; it does not identify which function or mechanism generates it, so applied work should pair the test with an estimator of the spillover function.","Because the composite s-test unions three hypotheses, its overall size may be larger than nominal; a researcher could apply a multiple-testing correction when using the rule of thumb.","The paper's simulations cover only size, not power; a natural next step is a power experiment with a known nonlinear spillover function (e.g., a quadratic) to verify the consistency claim in finite samples.","In the empirical y and cy tests, the instrument set is not reported; a careful replication should state the instruments used for the endogenous peer-outcome term, because the null distribution relies on their validity."],"forward_implications":["Applied researchers can test for nonlinear cross-unit spillovers using only the residuals from a linear IV/OLS regression under the null; no nonparametric estimation is required.","The test covers block-based and link-based interaction structures, extends to multiple weight matrices, and, under high-level conditions, tolerates noisy or incomplete network data.","The composite rule-of-thumb 's test' rejects if any of the c, y, or cy component tests rejects, which guards against masking when one channel is strong, the other weak, or when the series terms are highly collinear.","In four empirical settings (golf tournaments, university roommates, school deskmates, academic researchers), the test rejects the null of no spillovers in many specifications where linear peer-effect regressions do not reject.","For small samples, using χ²_q critical values rather than the limiting normal can improve size control, as shown in the paper's simulations."],"fun_headline_variants":["Nonparametric spillover test needs only a linear regression","Detect nonlinear cross-unit spillovers with one simple test","Spillovers via attributes or outcomes? Test uses standard normal","Test for spillovers that linear models often overlook"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For the outcome-channel tests, the null distribution of S relies on valid instruments for the endogenous peer-outcome term w_i'y: the orthogonality condition E(ε_i z_j)=0 must hold for every instrument z_j, and the four empirical applications never state which instruments are used for the y and cy tests, so those rejections are credible only if such instruments exist.","fun_headline_variants_meta":{"raw":{"variants":["Nonparametric spillover test needs only a linear regression","Detect nonlinear cross-unit spillovers with one simple test","Spillovers via attributes or outcomes? Test uses standard normal","Test for spillovers that linear models often overlook"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1178,"prompt_tokens":625,"completion_tokens":553,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":369,"completion_tokens_details":{"reasoning_tokens":488}},"tokens_in":369,"tokens_out":553,"duration_ms":6990,"temperature":1.0,"reasoning_tokens":488,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T01:12:05.366999+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run two simulations: (1) under the null of no spillovers but with an instrument that is correlated with a cluster-level shock, check whether the rejection rate of S stays near 5%; (2) under a known nonlinear spillover f(s)=s² with valid instruments, check whether power rises toward 1 with n. The paper reports only a size simulation with valid instruments, so neither of these checks is documented.","supporting_citations":[],"review_version":1}