An optimal test for strategic interaction in social and economic network formation between heterogeneous agents

We introduce a test for whether agents' preferences over network structure are interdependent. Interdependent preferences induce strategic behavior since the optimal set of links directed by agent i will vary with the configuration of links directed by other agents. Our model also incorporates agent-specific in- and out-degree heterogeneity and homophily on observable agent attributes. This introduces 2N + K2 nuisance parameters (N is number of agents in the network and K the number of possible agent attribute configurations). Under the null equilibrium is unique, but our hypothesis is nevertheless a composite one as the degree heterogeneity and homophily nuisance parameters may range freely across their parameter space. Under the alternative our model is incomplete; there may be multiple equilibrium network configurations and our test is agnostic about which one is selected. Motivated by size control, and exploiting the exponential family structure of our model under the null, we restrict ourselves to conditional tests. We characterize the exact null distribution of a family of conditional tests and introduce a novel Markov Chain Monte Carlo (MCMC) algorithm for simulating this distribution. We also characterize the locally best test. The form of this test depends upon the gradient of the likelihood with respect to the strategic interaction parameter in the neighborhood of the null. Remarkably, this gradient, and consequently the form of the locally best test statistic, does not depend on how an equilibrium is selected. Exploiting this lack of dependence, we outline a feasible version of the locally best test. We present two illustrative applications. First, we test for whether nations behave strategically when choosing locations for overseas diplomatic missions. Second, we test for whether firms prefer to sell to firms with richer customer bases (i.e., whether firms value "indirect customers"). Some Monte Carlo experiments explore the size and power properties of our test in practice.