Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Identification and Estimation of Network Models with Nonparametric Unobserved Heterogeneity

Andrei Zeleneev

TL;DR

This paper develops a semiparametric network model that allows nonparametric unobserved heterogeneity in dyadic interactions. By introducing a latent pseudo-distance $d_{ij}^2$ that captures similarity in fixed effects, it identifies and consistently estimates the covariate effect $\beta_0$ without specifying the dimensions or form of latent heterogeneity. It leverages latent-space/matrix-denoising techniques to identify and uniformly estimate the error-free outcome $Y_{ij}^*$ and the pair-specific fixed effects $g_{ij}$, enabling robust estimation via kernel-weighted pairwise differences. The framework extends to single-index and nonparametric models, accommodates missing outcomes, and scales to directed two-way networks, with extensive theoretical guarantees and numerical evidence showing substantial improvements over standard fixed-effects approaches in the presence of latent homophily.

Abstract

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.

Identification and Estimation of Network Models with Nonparametric Unobserved Heterogeneity

TL;DR

This paper develops a semiparametric network model that allows nonparametric unobserved heterogeneity in dyadic interactions. By introducing a latent pseudo-distance that captures similarity in fixed effects, it identifies and consistently estimates the covariate effect without specifying the dimensions or form of latent heterogeneity. It leverages latent-space/matrix-denoising techniques to identify and uniformly estimate the error-free outcome and the pair-specific fixed effects , enabling robust estimation via kernel-weighted pairwise differences. The framework extends to single-index and nonparametric models, accommodates missing outcomes, and scales to directed two-way networks, with extensive theoretical guarantees and numerical evidence showing substantial improvements over standard fixed-effects approaches in the presence of latent homophily.

Abstract

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.
Paper Structure (47 sections, 20 theorems, 152 equations, 2 tables)

This paper contains 47 sections, 20 theorems, 152 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Identification of the Semiparametric Model
  3. The model
  4. Identification of $\beta_0$: main insights
  5. Comparison with the existing approaches to identification
  6. Identification under conditional homoskedasticity
  7. Identification under general heteroskedasticity
  8. Identification of $Y_{ij}^*$
  9. Estimation of the Semiparametric Model
  10. Estimation of $\beta_0$
  11. Estimation of $d_{ij}^2$
  12. Estimation of $d_{ij}^2$ under conditional homoskedasticity of $\varepsilon_{ij}$
  13. Estimation of $d_{ij}^2$ under general heteroskedasticity of $\varepsilon_{ij}$
  14. Large Sample Theory
  15. Rate of convergence for $\hat{\beta}$
  16. ...and 32 more sections

Key Result

Theorem 1

Suppose that eq: R_n def holds for some $R_n \rightarrow \infty$. Then, under Assumptions ass: DGP-ass: local smooth g,

Theorems & Definitions (52)

  • Example 1: Nonparametric homophily based on unobservables
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Example : Illustration of Assumption \ref{['ass: xi dist']}\ref{['item: xi cont']}
  • Theorem 1
  • Remark 4.1
  • ...and 42 more