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Tractable Identification of Strategic Network Formation Models with Unobserved Heterogeneity

Wayne Yuan Gao, Ming Li, Zhengyan Xu

TL;DR

A tractable identification approach for strategic network formation models with both strategic link interdependence and individual unobserved heterogeneity (fixed effects) is developed using a ``bounding-by-$c$''technique that treats endogenous covariates as random variables and exploits monotonicity restrictions to obtain identifying information.

Abstract

We develop a tractable identification approach for strategic network formation models with both strategic link interdependence and individual unobserved heterogeneity (fixed effects). The key challenge is that endogenous network statistics (e.g. number of common friends) enter the link formation equation, while the mapping from model primitives to equilibrium network structure is generally intractable. Our approach sidesteps this difficulty using a ``bounding-by-$c$'' technique that treats endogenous covariates as random variables and exploits monotonicity restrictions to obtain identifying information. We derive a system of identifying restrictions based on subnetwork configurations: tetrad-based restrictions that completely eliminate all individual fixed effects, triad-based restrictions that partially difference out fixed effects, and general weighted cycle-based restrictions, along with point identification results. Preliminary simulations show that our approach can deliver informative bounds on the structural parameters.

Tractable Identification of Strategic Network Formation Models with Unobserved Heterogeneity

TL;DR

A tractable identification approach for strategic network formation models with both strategic link interdependence and individual unobserved heterogeneity (fixed effects) is developed using a ``bounding-by-''technique that treats endogenous covariates as random variables and exploits monotonicity restrictions to obtain identifying information.

Abstract

We develop a tractable identification approach for strategic network formation models with both strategic link interdependence and individual unobserved heterogeneity (fixed effects). The key challenge is that endogenous network statistics (e.g. number of common friends) enter the link formation equation, while the mapping from model primitives to equilibrium network structure is generally intractable. Our approach sidesteps this difficulty using a ``bounding-by-'' technique that treats endogenous covariates as random variables and exploits monotonicity restrictions to obtain identifying information. We derive a system of identifying restrictions based on subnetwork configurations: tetrad-based restrictions that completely eliminate all individual fixed effects, triad-based restrictions that partially difference out fixed effects, and general weighted cycle-based restrictions, along with point identification results. Preliminary simulations show that our approach can deliver informative bounds on the structural parameters.
Paper Structure (31 sections, 11 theorems, 81 equations, 1 figure, 2 tables)

This paper contains 31 sections, 11 theorems, 81 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Model and Assumptions
  3. Model Setup
  4. Assumptions
  5. Identifying Restrictions
  6. Tetrad-Based Restrictions
  7. Triad-Based Restrictions
  8. Weighted Differencing
  9. General Weighted Cycle-Based Differencing
  10. Primitive Conditions for Assumption \ref{['ass:tetrad_id']}
  11. Point Identification
  12. Setup and Definitions
  13. Tetrad Isolation
  14. Point Identification without Fixed Effects
  15. Point Identification with Fixed Effects
  16. ...and 16 more sections

Key Result

Theorem 1

Under Assumptions ass:random_sampling--ass:tetrad_id, the true parameter $\theta_{0}$ belongs to the identified set where the supremum and infimum are taken over $\zeta$ in the support of $\zeta_{ijhk}$. Equivalently, define Then $\Theta^{\mathrm{tetrad}}_{I}=\{\theta:Q^{\mathrm{tetrad}}(\theta)\leq0\}$.

Figures (1)

  • Figure 1: Baseline criterion $Q$ as a function of $\gamma$.

Theorems & Definitions (29)

  • Remark 1: Individual-Level versus Dyadic Rescaling
  • Remark 2: Population Identification versus Single-Network Estimation
  • Theorem 1: Tetrad Restrictions with Nonparametric $F_\Delta$
  • Remark 3: Outer Region
  • Theorem 2: Tetrad Restrictions with Parametric $F_{\Delta}$
  • Proposition 1: Three-Link Triad Bounds
  • Proposition 2: Two-Link Triad Bounds
  • Proposition 3: Weighted Differencing Bound
  • Definition 1: Weighted Link Configuration
  • Proposition 4: General Weighted Cycle-Based Identifying Restrictions
  • ...and 19 more