NBER WORKING PAPER SERIES AN OPTIMAL TEST FOR STRATEGIC INTERACTION IN SOCIAL AND ECONOMIC NETWORK FORMATION BETWEEN HETEROGENEOUS AGENTS

Consider a setting where N players, partitioned into K observable types, form a directed network. Agents' preferences over the form of the network consist of an arbitrary network benefit function (e.g., agents may have preferences over their network centrality) and a private component which is additively separable in own links. This latter component allows for unobserved heterogeneity in the costs of sending and receiving links across agents (respectively outand indegree heterogeneity) as well as homophily/heterophily across the K types of agents. In contrast, the network benefit function allows agents' preferences over links to vary with the presence or absence of links elsewhere in the network (and hence with the link formation behavior of their peers). In the null model which excludes the network benefit function, links form independently across dyads in the manner described by Charbonneau (2017). Under the alternative there is interdependence across linking decisions (i.e., strategic interaction). We show how to test the null with power optimized in specific directions. These alternative directions include many common models of strategic network formation (e.g., “connections” models, “structural hole” models etc.). Our random utility specification induces an exponential family structure under the null which we exploit to construct a similar test which exactly controls size (despite the null being a composite one with many nuisance parameters). We further show how to construct locally best tests for specific alternatives without making any assumptions about equilibrium selection. To make our tests feasible we introduce a new MCMC algorithm for simulating the null distributions of our test statistics. Andrin Pelican Rofna 7418 Domleschg Switzerland pelicanandrin@gmail.com Bryan S. Graham University of California Berkeley 530 Evans Hall #3880 Berkeley, CA 94720-3880 and NBER bgraham@econ.berkeley. edu In an economic model of (directed) network formation agents purposefully direct links to one another in order to maximize utility. Specifically, a payoff function maps all possible network configurations into agent utilities. Agents use this payoff function to weigh the benefits of directing any particular link against the costs of doing so. A Nash Equilibrium (NE) network arises when all agents link choices are individually optimal given the choices made by other agents (e.g. Bala and Goyal, 2000). Important examples of such processes include firms choosing their suppliers (e.g., Atalay et al., 2011), adolescents choosing friends (e.g., Christakis et al., 2020), banks engaging in interbank lending to meet statutory reserve requirements (e.g., Boss et al., 2004), and village households choosing partners for risk-sharing (e.g., De Weerdt, 2004). Jackson et al. (2017) present many other examples of networks in economics. Such data abound in the other social sciences as well (e.g., Apicella et al., 2012). The utility an agent receives when she directs a link to another agent can be usefully divided into two components. The first component is “private”, or, more precisely, invariant to the presence or absence of other links in the network. The second component is “social”, or varying with the presence or absence of other links in the network. An example of the first component is the payoff associated with a homophilous link (McPherson et al., 2001). This payoff component only depends on the attributes of the sending (ego) and receiving (alter) agents. Another example is associated with “degree heterogeneity”: agents may vary systematically in their propensity to direct links, or in their attractiveness as link targets for others. Finally we might posit that the payoff from any particular link varies for idiosyncratic reasons, as in other random utility models (RUMs) of discrete choice (McFadden, 1974). Empirical models of network formation with these features were introduced by Charbonneau (2017), Graham (2017), Dzemski (2018), Jochmans (2018) and Yan et al. (2018). These models are fundamentally dyadic: agents’ network payoffs are a simple sum of link-specific payoffs and, crucially, invariant to the linking behavior of other agents. In some settings, however, agents may also value indirect links. For example, an arc from j to k may incidentally reduce the shortest path length from i to k, allowing agent i better access to k’s information (e.g., Jackson and Wolinsky, 1996; Bala and Goyal, 2000). While arc jk is valued by i, this value is not incorporated into j’s decision to direct the arc or not. Preferences of this type mean agents’ decisions impose externalities on others. The detection of such externalities is the subject of this paper. In digraphs, or directed networks, it is customary to refer to edges as “arcs”. Here we use the terms link, edge, arc, friendship, relationship etc. interchangeably. Note “other links” include those possibility directed by the sending agent to targets other than the one at hand. An alternative to the “private” nomenclature would be “dyadic” or “direct”.