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We consider an index model of dyadic link formation with a homophily effectindex and a degree heterogeneity index. We provide nonparametric identificationresults in a single large network setting for the potentially nonparametrichomophily effect function, the realizations of unobserved individual fixedeffects and the unknown distribution of idiosyncratic pairwise shocks, up tonormalization, for each possible true value of the unknown parameters. Wepropose a novel form of scale normalization on an arbitrary interquantilerange, which is not only theoretically robust but also proves particularlyconvenient for the identification analysis, as quantiles provide directlinkages between the observable conditional probabilities and the unknown indexvalues. We then use an inductive "in-fill and out-expansion" algorithm toestablish our main results, and consider extensions to more general settingsthat allow nonseparable dependence between homophily and degree heterogeneity,as well as certain extents of network sparsity and weaker assumptions on thesupport of unobserved heterogeneity. As a byproduct, we also propose a conceptcalled "modeling equivalence" as a refinement of "observational equivalence",and use it to provide a formal discussion about normalization, identificationand their interplay with counterfactuals.

Nonparametric Identification in Index Models of Link Formation