Identification and Estimation of Network Models with Nonparametric Unobserved Heterogeneity

Andrei Zeleneev

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Identification and Estimation of Network Models with Nonparametric Unobserved Heterogeneity

Andrei Zeleneev

arXiv:2602.06885v1Feb 6, 2026econ.EM

TL;DR

The paper addresses identification when network outcomes reflect flexible, nonparametric unobserved heterogeneity and latent homophily. It identifies same-fixed-effect agents from interaction outcomes, uses their observed-characteristic variation to identify covariate effects, and develops consistent estimators supported by numerical illustrations.

Problem
Existing additive fixed-effects models cannot flexibly capture complex unobserved heterogeneity, while latent homophily may be correlated with observed characteristics in networks.
Method
The paper models fixed effects through an unrestricted smooth coupling function and identifies agents sharing fixed effects using an estimable pseudo-distance from interaction outcomes.
Results
The proposed identification strategy recovers covariate effects while controlling for fixed effects, identifies error-free outcomes and pair-specific fixed effects, and supports consistent estimators.
Takeaways & Limitations
Interaction outcomes can reveal latent fixed-effect groupings, enabling identification of covariate effects in network models with flexible unobserved heterogeneity.
Takeaways & Limitations
Matching agents with identical observed characteristics and fixed effects alone cannot disentangle observable and unobservable effects to identify β0.

Abstract

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Homophily based on observables is widespread in networks. Therefore, homophily based on unobservables (fixed effects) is also likely to be an important determinant of the interaction outcomes. Failing to properly account for latent homophily (and other complex forms of unobserved heterogeneity) can result in inconsistent estimators and misleading policy implications. To address this concern, we consider a network model with nonparametric unobserved heterogeneity, leaving the role of the fixed effects unspecified. We argue that the interaction outcomes can be used to identify agents with the same values of the fixed effects. The variation in the observed characteristics of such agents allows us to identify the effects of the covariates, while controlling for the fixed effects. Building on these ideas, we construct several estimators of the parameters of interest and characterize their large sample properties. Numerical experiments illustrate the usefulness of the suggested approaches and support the asymptotic theory.

1 Introduction

The paper develops a network model with flexible nonparametric unobserved heterogeneity, motivated by latent homophily and limitations of additive or low-rank approaches. It uses interaction outcomes to identify agents sharing fixed effects, enabling estimation of covariate effects and pair-specific heterogeneity.

Motivation: Latent homophily may importantly shape network interactions because observed and unobserved characteristics are typically correlated.The paper emphasizes that additive fixed effects may not capture more complicated forms of unobserved heterogeneity.
Model: The model leaves the coupling function, fixed-effect dimension, and fixed-effect interactions unspecified, while allowing nonlinear network and Poisson specifications through a known invertible link.This framework generalizes additive fixed effects and accommodates homophily based on unobservables.
Identification: The paper argues that interaction outcomes identify agents with equal fixed effects through a pseudo-distance that is zero if and only if their fixed effects coincide.The pseudo-distance is identified and can be estimated from the data.
Identification: Variation in observed characteristics among agents with identical fixed effects identifies the covariate parameter while controlling for unobserved heterogeneity.The argument also applies when the observable component is an unknown nonparametric function.
Estimation: The approach estimates error-free outcomes under homoskedastic and general heteroskedastic errors, including uniformly consistent recovery across all agent pairs.The heteroskedastic case builds on and extends matrix estimation and completion methods.
Implications: Identifying pair-specific fixed effects supports policy-relevant pair-specific and average partial effects when the link function is nonlinear.The paper presents the error-free-outcome result as applicable beyond the specific model and as a foundation for further identification results.
Relation to existing approaches: The paper contrasts its flexible framework with low-rank approximations and clustering methods that impose smoothness, high-rank, discreteness, or informative individual-specific-moment requirements.These restrictions can be problematic in network settings where informative individual-specific moments may not exist.

2 Identification of the Semiparametric Model

The paper identifies covariate effects in network models with unrestricted unobserved heterogeneity by using interaction outcomes to find agents sharing fixed effects but differing in observables.

Model: The model allows unknown coupling between multidimensional unobserved fixed effects and accommodates undirected networks, with extensions to directed and two-way settings.The observed covariates enter through a known transformation, while the coupling function and fixed-effect dimension remain unrestricted.
Identification: Agents with identical unobserved characteristics can be identified using pseudo-distances constructed from interaction outcomes, including under general heteroskedasticity.The approach identifies error-free outcomes uniformly across agent pairs, enabling pseudo-distance-based matching.
Identification: Matching agents with the same fixed effects but different observed characteristics preserves variation needed to identify β0 while controlling for unobserved heterogeneity.The method avoids relying on high-rank pair-specific exogenous variation required by some panel approaches.
Comparison with existing approaches: The proposed matching strategy differs from existing clustering and similarity methods because those methods may discard observed characteristics or match on both observed and unobserved characteristics.Matching on both components leaves no identifying variation for separating observable and unobservable effects.
Comparison with existing approaches: The approach also addresses limitations of individual-specific-moment clustering, which can be uninformative under symmetric latent homophily structures.In the spherical example, all such moments fail to contain information about an agent’s latent characteristic.
Nonparametric extension: The identification of error-free outcomes extends beyond the semiparametric specification to fully non-separable nonparametric network models.Treating error-free outcomes as observed simplifies further identification analysis in more complicated models.

3 Estimation of the Semiparametric Model

The section develops estimators for the pseudo-distances that reveal latent similarity and uses them to estimate β0 through weighted pairwise-difference regressions. It also constructs estimators of the error-free outcomes under homoskedastic and heteroskedastic errors.

Estimating pseudo-distances: The estimation procedure first constructs pseudo-distance estimates and then uses them to identify agents with similar latent characteristics.These estimates are developed for both homoskedastic and general heteroskedastic settings.
Estimating β0: The proposed β̂ combines pairwise-difference regressions weighted toward pairs whose estimated pseudo-distance is small.The kernel bandwidth shrinks with sample size, improving match quality and reducing imperfect-matching bias.
Estimating pseudo-distances: Under homoskedasticity, the pseudo-distance is estimated from a related observable quantity after accounting for the conditional error variance.The construction uses the minimum of a pairwise quantity to estimate the variance component.
Estimating error-free outcomes: Under general heteroskedasticity, the procedure constructs neighborhoods of agents with identical observed characteristics and similar estimated latent characteristics.These neighborhoods support estimation of the error-free outcomes, including outcomes for unobserved diagonal pairs.
Estimating error-free outcomes: The error-free-outcome estimator extends graphon-estimation ideas and is presented as a matrix estimation or completion problem with formal statistical guarantees.Candidate estimators can be chosen according to the setting, with the paper extending Zhang et al. (2017)’s approach.

4 Large Sample Theory

The large-sample theory establishes consistency and convergence rates for β̂ and the error-free-outcome estimator under regularity and identification conditions. It also establishes identification of pair-specific fixed effects and extends guarantees to observed covariates and latent dimensions beyond earlier settings.

Convergence of β̂: The estimator is consistent when the latent effect is discrete, and it becomes asymptotically equivalent to an oracle estimator using true cluster membership.In this case, β̂ is asymptotically normal and unbiased.
Convergence of β̂: Theorem 1 establishes a guaranteed convergence rate for β̂ under the paper’s regularity conditions and a high-level rate condition for estimated pseudo-distances.The paper emphasizes consistency and identification rather than asymptotic normality in the general setting.
Convergence of β̂: The candidate pseudo-distance estimators can satisfy the required rate condition with R_n=(n/ln n)^1/2, or a slower-growing rate depending on latent dimension.Under such rates, the general convergence bound simplifies.
Scope of inference: The paper’s general theory focuses on consistency because obtaining asymptotic normality or inference would require additional assumptions considered restrictive for network models.The authors characterize this choice as a consequence of the model’s representative but nonstandard features.
Estimating error-free outcomes: Theorem 2 establishes asymptotic properties for the error-free-outcome estimator using neighborhoods whose sizes grow at a prescribed rate.The result extends earlier guarantees to higher-dimensional latent effects, non-binary outcomes, unbounded errors, and discrete observed covariates.
Continuous observed covariates: The framework can consistently estimate β0 when observed covariates are continuously distributed by treating them as part of the effective latent variable.The resulting rate depends on the combined dimension of ξ and X.
Identification results: The paper establishes uniform consistency and identification of the error-free outcomes Y* and identifies pair-specific fixed effects g(ξ_i,ξ_j).The pair-specific fixed effects are consistently estimated for every agent pair when β̂ is consistent.

5 Extensions

The paper extends its identification strategy from semiparametric network models to nonlinear, nonparametric, and directed settings. It identifies observable effects, pair-specific fixed effects, and policy-relevant partial effects under broader structures.

The framework extends identification arguments to a wide range of semiparametric and nonparametric network models beyond the baseline specification.
General nonlinear models: In the general model Yij = f(Xi, ξi, Xj, ξj) + εij, error-free outcomes are identified even when the structural function f itself is not.The paper stresses that identifying f requires additional structure because ξ is unobserved.
General nonlinear models: Because the link function is invertible, the approach identifies β0 and all pair-specific fixed effects in nonlinear network models.
General nonlinear models: Identifying pair-specific fixed effects allows identification of pair-specific and average partial effects, even when β0 alone is insufficient because F(·) may be nonlinear.
Nonparametric model: In the nonparametric additive model, h(·, ·) and pair-specific fixed effects are nonparametrically identified, so earlier results do not depend on linearity in observables.
Nonparametric model: The nonparametric model is overidentified and can, in principle, be falsified because the relevant pseudo-distance should agree across agents with matching fixed effects.
Further extensions: The estimation procedure can also be generalized to settings with missing interaction outcomes, directed networks, and more general two-way models.

6 Numerical Evidence

Numerical experiments show that controlling for nonparametric unobserved heterogeneity improves estimation relative to additive fixed-effects or naive approaches. The empirical illustration and calibrated simulation indicate that standard methods can overestimate homophily.

Homoskedastic simulation: The proposed kernel and nearest-neighbor estimators remove bias in moderate networks, while the naive fixed-effects estimator becomes biased when observed and unobserved characteristics correlate.
Homoskedastic simulation: When fixed effects contribute substantially to outcome variability, differencing them out can improve precision despite reducing the effective sample size.
Homoskedastic simulation: The simulation compares additive fixed effects, kernel, and 1-nearest-neighbor estimators across network sizes and correlations between observed and unobserved characteristics.
Empirical illustration: The Facebook 100 application studies gender homophily using friendship networks and nodal covariates from 100 U.S. colleges and universities.
Empirical illustration: In the calibrated experiment, naive maximum-likelihood and tetra-logit estimators severely overestimate β0, whereas the proposed estimator correctly estimates the true effect with comparable standard deviation.
Empirical illustration: The calibrated results support the empirical finding that standard approaches overestimate gender homophily, with differences unlikely to be entirely due to sampling variability.

A.1 Proofs of the results of Section 4.1

The appendix proves that kernel weighting excludes sufficiently dissimilar fixed effects asymptotically and controls the contribution of nearby pairs. These bounds support the estimator’s asymptotic analysis.

For sufficiently large networks, pairs whose fixed effects differ by more than αh_n receive zero kernel weight with probability approaching one.
The pseudo-distance is bounded below by a multiple of squared fixed-effect differences near equality, separating sufficiently dissimilar agents from nearby pairs.
The expected number of agents within an αh_n neighborhood is uniformly bounded, supporting control of the number of effectively matched pairs.

A.1.2 Proof of Theorem 1

The proof of Theorem 1 establishes convergence of the estimator’s matrix components using boundedness, kernel localization, dominated convergence, and U-statistic arguments. These results yield the required positive-definite limiting matrix.

The proof decomposes the estimator into Â_n, B̂_n, and Ĉ_n and establishes convergence rates for these components.
The limiting matrix satisfies λmin(A) > C for some C > 0, completing the positive-definiteness step required by the theorem.
For dimensions d_ξ > 1, the proof remains essentially unchanged after normalizing the relevant quantities by h_n.
The matrix Â_n converges to a symmetric invertible matrix A through kernel bounds, dominated convergence, and U-statistic approximation.
Uniform boundedness of the kernel terms and conditional expectations permits dominated convergence in evaluating the limiting matrix.

A.1.4 Bounding ˆBn

This section bounds the components entering ˆB_n and establishes the stochastic orders needed for its control. The proof combines conditional kernel bounds, U-statistic inequalities, and uniform consistency arguments.

Bounding b_n: b_n := E[q_n(Z_i, Z_j, Z_k)] = O(h_n^2).This bound is obtained from conditional expectations and continuity conditions, with exceptional regions contributing only O(h_n).
Bounding B_n: B_n is a third-order U-statistic with symmetrized kernel p_n and expectation b_n.The proof then applies a Bernstein-type inequality for U-statistics to control B_n.
Bounding ˆB_n: The estimator difference ˆB_n − B_n is bounded using Lemma A.1(ii), Assumption 5(i), and a term of order O_p(h_n).Together with the bound on B_n, these ingredients deliver the required result for ˆB_n.
Uniform consistency: ˆω_n,ik is uniformly consistent for ω_n,ik, and its norms are bounded with probability approaching one.Uniform consistency is established through the analogous consistency of ˆκ_n,ik and η_n,ik, followed by boundedness arguments.
Bounding C_n: C_n = O_p(n^-1), obtained by applying concentration bounds to independent weighted error vectors after controlling max_i≠k ∥ˆω_n,ik∥.The final bound uses the high-probability boundedness of the estimated weights and Theorem A.2.

A.2 Proof of the results of Section 4.2.1

This section proves auxiliary uniform bounds under compactness and the stated assumptions. The results control the relevant distance and covariance quantities used in Section 4.2.1.

Auxiliary bounds: Under Assumptions 1 and 2 with compact B, the auxiliary quantities in Lemma A.4 admit uniform stochastic bounds.The proof uses boundedness, conditional concentration, and a union bound over agent pairs.
Estimated differences: Uniform control of estimated differences follows by combining bounds on ˆ∆_ij(β) with concentration for sub-Gaussian error differences.Uniformity over agent pairs is ensured by Assumption 2(iv) and a union bound.
Conclusion: The resulting bounds imply the stated maximal convergence result for the relevant pairwise quantities.The proof concludes the first part of the lemma after combining the two uniform components.

A.3.1 Proof of Theorem 2

The proof of Theorem 2 establishes that sufficiently many nearby agents exist and uses them to control approximation errors uniformly across agents and neighboring pairs.

Nearby-agent counts: For δ_n,C = a_C(n^-1 ln n)^(1/(2d_ξ)), every agent has at least C(n ln n)^(1/2) nearby agents with probability approaching one.This follows from concentration for neighborhood counts and the lower bound on local probability mass.
Neighbor bounds: Lemma A.6 supplies the bounds needed for neighboring-agent quantities, including cases where the indexing agent coincides with one comparison partner.The proof treats distinct and coinciding index cases separately before completing each theorem component.
Pairwise comparisons: The proof uses these nearby agents to compare agents i and i′ in the estimated interaction outcomes.Boundedness of Y* and Assumption 2(iii) control the comparison uniformly over the relevant agent pairs.
Concentration: Bernstein inequalities and union bounds provide the uniform concentration steps used throughout the proof of Theorem 2.These arguments control the decomposed terms S_i and S_2i and deliver the desired results for both theorem parts.

A.3.2 Proof of Theorem 3

The proof of Theorem 3 controls the estimated pairwise outcomes and projection terms using compactness, boundedness, and the results established for Theorem 2.

Coefficient control: Lemma A.7 establishes a compact parameter set containing the estimated pairwise coefficients with probability approaching one.The argument uses finite support for X, a uniformly positive minimal nonzero eigenvalue, and boundedness of the relevant quantities.
Coefficient estimation: The Moore–Penrose inverse is used to characterize the minimum in the pairwise coefficient estimation problem.The resulting coefficient estimates are shown to be uniformly bounded over agent pairs.
Proof reduction: Theorem 3 represents the estimation error through ˆ∆Y*_{i−j} and the projection matrix P_{i−j}.The proof reduces the target bounds to controlling normalized inner products involving the estimated and true pairwise outcomes.
Use of Theorem 2: The required bounds follow by invoking Theorem 2(ii) and Theorem 2(i) for the corresponding error terms.These two results complete the final convergence steps in the proof of Theorem 3.

A.4 Proof of Theorem 4

The proof establishes the theorem by controlling approximation errors uniformly over candidate matches and applying concentration bounds with a union bound. It also illustrates how the smoothness requirement accommodates a non-differentiable interaction function.

The proof reduces the result to showing that two maximum approximation-error terms are op(1).
Uniform closeness of latent characteristics within matched neighborhoods, together with Assumption 2(iii), controls differences in g across matched and original agents.
Conditional Bernstein inequalities and uniform sub-Gaussianity bound deviations over every admissible matching realization, after which a union bound controls their aggregate probability.
The proof concludes because the logarithmic complexity term is smaller than Cn^1/2(ln n)^3/2 and m exceeds a constant multiple of n ln n.
For g(ξi, ξk) = κ|ξi −ξk|, the linearization remainder need not be O(|ξi −ξj|^2), but Assumption 6 still bounds it under the stated separation restrictions.

C Imputation of ˆY ∗ ij with Missing Data

This section extends imputation of missing interaction outcomes by constructing donor neighborhoods from observed matches and allowing sequential expansion when the original donor pool is too sparse. The resulting imputed outcomes support estimation of distance measures and the parameter estimator.

For d̂2∞(i,j), agents k with too few common observed neighbors can be dropped, provided the remaining comparison subset is sufficiently large.
Candidate donors for imputing Y*ij match agent i on X and have observed outcomes Yi′j.
The donor neighborhood N̂ij(nij) contains nij agents from Pij chosen as closest to i according to d̂2ij.
If the observed matrix is too sparse to provide enough donors, newly imputed outcomes can sequentially expand the pool of potential donors.
After constructing Ŷ*, the procedure estimates d̂2ij and then calculates β̂, optionally using imputed outcomes in the estimator and overlap definition.