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Empirical Challenges with Peers-of-Peers Instruments in the Linear-In-Means Model

Nathan Canen, Shantanu Chadha

Abstract

In the linear-in-means model, endogeneity arises naturally due to the reflection problem. A common solution is to use Instrumental Variables (IVs) based on higher-order network links, such as using friends-of-friends' characteristics. We first show that such instruments are unlikely to work well in many applied settings: in very sparse or very dense networks, friends-of-friends may be similar to the original links. This implies that the IVs may be weak or their first stage estimand may be undefined. For a class of random graphs, we use random graph theory and characterize regimes where such instruments perform well, and when they would not. We prove how weak-IV robust inference can be adapted to this environment, and how scaling the network can help. We provide extensive Monte Carlo simulations and revisit empirical applications, showing the prevalence of such issues in empirical practice, and how our results restore valid inference.

Empirical Challenges with Peers-of-Peers Instruments in the Linear-In-Means Model

Abstract

In the linear-in-means model, endogeneity arises naturally due to the reflection problem. A common solution is to use Instrumental Variables (IVs) based on higher-order network links, such as using friends-of-friends' characteristics. We first show that such instruments are unlikely to work well in many applied settings: in very sparse or very dense networks, friends-of-friends may be similar to the original links. This implies that the IVs may be weak or their first stage estimand may be undefined. For a class of random graphs, we use random graph theory and characterize regimes where such instruments perform well, and when they would not. We prove how weak-IV robust inference can be adapted to this environment, and how scaling the network can help. We provide extensive Monte Carlo simulations and revisit empirical applications, showing the prevalence of such issues in empirical practice, and how our results restore valid inference.
Paper Structure (19 sections, 5 theorems, 93 equations, 12 figures, 11 tables)

This paper contains 19 sections, 5 theorems, 93 equations, 12 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Weak Instruments in the Linear-in-Means Model
  4. Model
  5. Identification
  6. Empirical Challenges with Network-Based Instruments
  7. Challenges with Peers-of-Peers Instruments in Random Graph Models
  8. Theoretical Results
  9. Sign reversals and near-boundary instability
  10. Simulations
  11. Weak-IV Robust Testing
  12. Simulations on Random Graphs
  13. Application of Weak IV Robust Inference with Network-Based Instruments
  14. Conclusion
  15. Appendix
  16. ...and 4 more sections

Key Result

Proposition 3.1

Suppose that Assumption 3.1 holds, that $\beta \boldsymbol{\gamma} + \boldsymbol{\delta} \neq 0$ and that the matrices $I, \mathbf{G}, \mathbf{G}^2$ are linearly independent. Then, the social effects $\mathbf{\theta}=(\alpha, \beta, \boldsymbol{\gamma}', \boldsymbol{\delta}')'$ are identified. If th

Figures (12)

  • Figure 1: US Congressional Alumni Network 109-113 (battaglini2018)
  • Figure 2: Network of Allies from the Second Congo War (Konig2017)
  • Figure 3: Network of Enemies from the Second Congo War (Konig2017)
  • Figure 4: Family Network from sample municipality from Philippines (Cruz2017)
  • Figure 5: Variance Normalized Partial-Covariance between the two Endogenous Variables with instruments (Konig2017)
  • ...and 7 more figures

Theorems & Definitions (13)

  • Proposition 3.1: bramoulle2009identification, Proposition 1
  • Definition 3.1: Weak identification in networks
  • Proposition 1: Upper bound on variance--normalized covariance in Erdős--Rényi graphs
  • proof
  • Proposition 2
  • proof
  • Lemma 1: Eigenvalue decomposition of the first-stage covariance
  • proof
  • Theorem : Theorem 1.1 of krivelevich2001
  • Definition 7.1: Spectral Norm
  • ...and 3 more