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REVIEW 2 major objections 5 minor 64 references

Estimating Peer Effects Using Partial Network Data

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Peer effects can be consistently estimated from partial network data whenever the network-formation parameters are identifiable, and correcting for missing links in Add Health raises the estimated effect by about 50%.

desk verdict A solid, useful paper on estimating peer effects with partial network data, but the main theorem assumes the key global-identification condition rather than proving it; that is the referee's main ask. read the letter →

arxiv 2509.08145 v1 pith:6ERNIRUR submitted 2025-09-09 econ.EM

Vincent Boucher , Aristide Houndetoungan This is my paper
classification econ.EM
keywords SocialnetworksPeereffectsMissingvariablesMeasurementerrorsSimulatedGMMBayesianestimationLinear-in-meansPartialnetworkdata
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether peer effects can still be estimated when the researcher does not see the whole social network—only a sample of links, a censored list, or a noisy version of the adjacency matrix. The central claim is yes, provided the partial data allow a consistent estimate of the distribution from which the true network was drawn. On that basis the paper constructs two estimators: a computationally cheap simulated GMM that corrects the bias from replacing the true network with simulated draws, and a fuller Bayesian estimator that treats the network as missing data. The empirical payoff is concrete: in the Add Health school data, roughly 45% of friendship nominations carry errors, and treating the observed network as true yields an endogenous peer effect about two-thirds the size of the estimate obtained on the reconstructed network.

What carries the argument

The load-bearing object is the estimated distribution of the true network, constructed from a network formation model estimated on partial data—sampled, censored, or misclassified links. The SGMM machinery is the bias-corrected simulated moment function (4), which uses three independent draws from that distribution: one draw builds the instruments, a second builds the regressors, and a third approximates the bias that would otherwise remain because the simulated network does not converge to the true network within bounded groups. The Bayesian machinery is data augmentation: the network itself is treated as an unknown parameter, updated by a Metropolis-Hastings step whose prior is the estimat

What would settle it

Take a fully observed network, artificially censor or misclassify links at known rates, and compare the SGMM estimate to the estimate on the complete network; if coverage of the true coefficient deteriorates sharply, or if the concentrated objective shows multiple minima on a grid of alpha values, the central claim fails.

Watch

Extended reading notes

Core claim

The paper's formal claim is Theorem 1: under exogeneity of the network, bounded independent groups, a parametric link-independent network formation model, and the existence of a √M-consistent estimator of the formation parameters from the partial data, the simulated GMM estimator defined by the moment function (4) is consistent and asymptotically normal for any positive numbers of simulated draws R, S, T. The moment function removes the approximation error caused by substituting simulated networks for the true one: instruments come from one set of draws, the regressors from a second, and a third draw approximates the asymptotic bias term, so that the moment is zero at the true parameters. Th

Load-bearing premise

The result stands on the premise that partial network data identify the network-formation parameters at the usual statistical convergence rate; the additional global identification condition (a unique minimum of the concentrated objective) is supported only by simulation evidence of convexity, not by a proof.

Editorial extensions

If this is right

  • A single cross-section of partial network data—sampled pairs, censored nominations, or misclassified links—can be enough to identify the endogenous peer effect, as long as the formation parameters are identified.
  • Standard instrumental-variable estimates that treat the observed network as complete are downward-biased; in Add Health, the bias is large enough that the estimated endogenous effect grows by about 50% after reconstruction.
  • The SGMM estimator stays centered on the true value in simulations even when half of the links are missing, with precision declining as missingness grows.
  • The Bayesian estimator remains valid in finite samples and can accommodate network formation models, such as those with unobserved degree heterogeneity or aggregated relational data, that fall outside the SGMM's asymptotic framework.
  • The reconstructed network changes which individuals would be chosen as 'key players' for interventions, not just the size of the social multiplier.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if this downward-bias pattern holds in other nomination-based surveys (many ask for a fixed number of friends), published peer-effect coefficients from such data are likely conservative, and re-estimating with reconstructed networks could change policy conclusions.
  • Beyond the paper: because the global identification condition is not proved, a practical robustness check is to sweep the concentrated objective over a grid of alpha values in each application and report whether a unique minimum is visible.
  • Beyond the paper: the framework suggests a testable extension—using the same reconstruction on data sets where the full network is known but deliberately noised, to quantify how misspecification of the network-formation model propagates into the peer-effect estimate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies estimation of linear-in-means peer effects when the true adjacency matrix is not fully observed. It proposes an SGMM estimator built from draws of an estimated network formation distribution and a Bayesian data-augmentation estimator. The main theoretical result (Theorem 1) states that, under high-level regularity and a global identification condition, the SGMM is consistent and asymptotically normal for any positive R,S,T. The paper reports simulations for missing and misclassified links and an Add Health application in which correcting for missing nominations raises the estimated endogenous peer effect by about 50%.

Significance. If the theorem were fully established, the paper would make a useful contribution: it unifies sampled, censored, and misclassified networks under one computational framework, avoids numerical integration over networks, and ships an R package. The bias-corrected moment construction is clever, and the treatment of first-stage uncertainty in the variance estimation is careful. The simulations are encouraging. However, the central result is conditional on an unproved global identification assumption, and the asymptotic normality proof has a joint-convergence gap. The empirical 1.5x claim is driven by the Bayesian estimator; the SGMM estimates are not statistically distinguishable. With a strengthened theorem for leading cases, the contribution would be solid.

major comments (2)
  1. [Theorem 1; Online Appendix C.1] The global identification condition in Theorem 1 is assumed, not derived. Lemma 1 establishes that E[m_m(θ0,ρ0)] = 0, but consistency requires θ0 to be the unique zero of lim E[mbar_M(θ,ρ0)]. Online Appendix C.1 reduces the condition to full rank of Bbar_0(α) plus unique minimization of the concentrated objective Q^c_0(α), and then states that 'simplifying this condition is challenging' and appeals to simulations showing strict convexity. This is not a proof, and nothing in Assumptions 1–5 rules out a second minimum or a flat region for some DGPs (e.g., dense or strongly clustered networks, or α near the boundary). Because the abstract claims that a consistent estimator of the network distribution is sufficient, the paper should either provide primitive identification conditions for the leading logistic-link cases or qualify the claim as conditional on global identification. The empirica
  2. [Appendix A.2, Eq. (8)] The asymptotic normality proof is incomplete. The text states that, conditional on sqrt(M)(rho_hat - rho_0), a Lyapunov CLT applies to sqrt(M) mbar_M(θ0,ρ0). But rho_hat depends on all M groups, so conditioning on sqrt(M)(rho_hat - rho_0) does not restore independence across the m-th moment terms; the conditioning event is a function of the entire sample. A rigorous joint convergence argument is needed, e.g., an influence-function expansion for rho_hat together with a Cramér-Wold argument for the vector of averages, or adding the first-stage moment conditions to the GMM system. As written, Theorem 1's asymptotic normality and the variance estimator in Online Appendix C.2 rest on an unproven joint distribution.
minor comments (5)
  1. [Abstract; Section 5] The abstract says network data errors have a large downward bias. In Tables G.2–G.3, the SGMM point estimate moves from 0.455 to 0.683–0.753, but the standard errors are about 0.23–0.25, so the difference is not statistically significant. Footnote 35 admits this only for the Bayesian estimator; the abstract and Section 5 should be qualified.
  2. [Table G.3] There are typos in the table: '0.03)' should be '0.03' and '0.08)' should be '0.08'. Similar small typos appear in the notes to Figure G.4.
  3. [Online Appendix C.1] The statement that Q^c_0(α) is 'strictly convex' in numerous simulation exercises would be more credible if a figure or grid evaluation of the concentrated objective were included, at least for one representative DGP.
  4. [Section 4] The claim that the Bayesian estimator is 'valid in finite samples' is imprecise: the MCMC targets the posterior distribution, but no posterior consistency or finite-sample risk statement is proved. Suggest rewording to 'has a finite-sample Bayesian interpretation'.
  5. [Section 5 and Table G.2] The text says about 45% of friendship nominations are coded with error, while Table G.2 reports 60% 'proportion of inferred network data'. The denominators differ (named nominations vs. all potential network entries) and should be reconciled explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the core SGMM argument is a standard two-step simulated-moment construction; the assumed global identification condition is an unproven limitation, not a circular reduction.

full rationale

The paper's central claim is that a consistent estimator of the network-formation distribution (Assumption 5) suffices, together with a standard global identification condition, for consistency and asymptotic normality of the SGMM estimator. Lemma 1 shows the moment function has zero mean at (θ0, ρ0) because the simulated draws and the true network share the same distribution at ρ0; this is the usual simulation-based moment validity argument, not a definitional identity. The identification condition (lim E(m̄_M(θ,ρ0)) ≠ 0 for θ≠θ0) is explicitly assumed in Theorem 1, and Online Appendix C.1 admits 'simplifying this condition is challenging' and offers only simulation evidence of convexity. That is a gap between the abstract's 'sufficient' claim and the theorem, but it is a correctness/identification concern, not circularity: the condition is not derived from the object it seeks to identify. The empirical application estimates the network-formation model from the observed partial network data and then uses draws from that estimated distribution to form instruments; this is a two-step simulated GMM, not a fitted parameter being renamed a prediction. The Bayesian estimator's use of the outcome in reconstructing the network is standard data augmentation, and the reported 1.5× ratio is a comparison of estimates, not a construction. The only self-citations (e.g., Houndetoungan and Maoude 2024 for variance estimation; Hsieh, Lee and Boucher 2020; Boucher and Mourifié 2017 for extensions) are technical or peripheral and are not load-bearing for the main theorems. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on standard econometric assumptions plus two non-standard ones: the high-level Assumption 5 and the unproven global identification condition. The empirical application further relies on a fitted logit network formation model and selected schools.

free parameters (3)
  • Network formation coefficients ρ = e.g., same sex 0.31, age diff -0.70 (Add Health)
    Fitted to the observed network in the application; the corrected peer effect estimates and the '1.5x larger' claim depend on this model.
  • Misclassification rates in simulations = false positive 0-15%, false negative 15-30%
    Assumed values in Monte Carlo; not the central claim.
  • Bayesian prior hyperparameters = μ_ᾶ=-1, σ_ᾶ^-2=2, a=b=4, Σ_Λ^-1=I/100
    Chosen, weakly informative priors for the Bayesian estimator.
assumptions (6)
  • domain assumption Assumption 1: M groups of bounded size, independent across m
    Provides many-market asymptotics; rules out unobserved degree heterogeneity that needs large groups.
  • domain assumption Assumption 4: E[ε_m | X_m, A_m, A_bar_m] = 0 (exogenous network)
    Benchmark assumption; the paper does not address network endogeneity.
  • domain assumption Assumption 5: there exists a sqrt(M)-consistent estimator of the network formation parameters from partial data
    Load-bearing premise for the entire method.
  • domain assumption Conditional link independence in network formation: P(A_m|X_m)=Π_ij P(a_ij,m|X_m)
    Used to construct the distribution of the true network; extensions to some ERGMs are discussed in the appendix.
  • ad hoc to paper Global identification: for any θ≠θ0, lim_{M→∞} E(¯m_M(θ,ρ0)) ≠ 0
    Assumed in Theorem 1; the paper cannot reduce it to a simple rank condition and only provides simulation evidence.
  • domain assumption For the Bayesian estimator: ε_m ~ N(0, σ^2 I) and priors on α, β, γ, σ^2
    Enables likelihood and MCMC; normality is a strong parametric assumption.

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Pith. "Pith review of Estimating Peer Effects Using Partial Network Data." pith.science (2026). https://pith.science/paper/6ERNIRUR

@misc{pith2026250908145,
  author       = {Pith},
  title        = {Pith review of: Estimating Peer Effects Using Partial Network Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ERNIRUR}},
  note         = {Machine review of arXiv:2509.08145}
}
read the original abstract

We study the estimation of peer effects through social networks when researchers do not observe the entire network structure. Special cases include sampled networks, censored networks, and misclassified links. We assume that researchers can obtain a consistent estimator of the distribution of the network. We show that this assumption is sufficient for estimating peer effects using a linear-in-means model. We provide an empirical application to the study of peer effects on students' academic achievement using the widely used Add Health database, and show that network data errors have a large downward bias on estimated peer effects.

Figures

Figures reproduced from arXiv: 2509.08145 by the authors.

Figure 1
Figure 1. Estimated peer effects with mismeasured links [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
Figure 2
Figure 2. Frequencies of the number of missing links per adolescent [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Peer effect estimate Note: Dots represent estimated values (and posterior mean) of α, and bars represent 95% confidence intervals (and 95% credibility intervals). Tables G.2 and G.3 in Online Appendix G.3 present the full set of estimated coefficients. We first see that the SGMM estimator is less efficient than the Bayesian estimator. This 28 [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

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