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

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.