Nonparametric Identification in Index Models of Link Formation

TL;DR

This paper establishes nonparametric identification of key components in a dyadic link formation model using a novel scale normalization and an inductive algorithm, accommodating complex heterogeneity and network sparsity.

Contribution

It introduces a new scale normalization method and an inductive algorithm for nonparametric identification in network models with unobserved heterogeneity.

Findings

Identification of homophily effect function and unobserved effects up to normalization

Development of a robust scale normalization based on interquantile range

Extension of identification results to more general network settings

Abstract

We consider an index model of dyadic link formation with a homophily effect index and a degree heterogeneity index. We provide nonparametric identification results in a single large network setting for the potentially nonparametric homophily effect function, the realizations of unobserved individual fixed effects and the unknown distribution of idiosyncratic pairwise shocks, up to normalization, for each possible true value of the unknown parameters. We propose a novel form of scale normalization on an arbitrary interquantile range, which is not only theoretically robust but also proves particularly convenient for the identification analysis, as quantiles provide direct linkages between the observable conditional probabilities and the unknown index values. We then use an inductive "in-fill and out-expansion" algorithm to establish our main results, and consider extensions to more general settings that allow nonseparable dependence between homophily and degree heterogeneity, as well as certain extents of network sparsity and weaker assumptions on the support of unobserved heterogeneity. As a byproduct, we also propose a concept called "modeling equivalence" as a refinement of "observational equivalence", and use it to provide a formal discussion about normalization, identification and their interplay with counterfactuals.