The Spatial Probit Model of Interdependent Binary Outcomes: Estimation, Interpretation, and Presentation

Interdependence—i.e., that the outcomes in or actions or choices of some units depend on those in/of others—is substantively and theoretically ubiquitous in and central to binary outcomes of interest across the social sciences. Most empirical applications omit interdependence, however; even theoretical and substantive discussion usually ignores it. Moreover, in the few contexts where spatial interdependence has been acknowledged or emphasized, such as in the social-network and policy-diffusion literatures, models do not fully reflect simultaneity of the outcomes across units and/or the endogeneity of the spatial lags (appropriately) employed to model the interdependence has gone unrecognized. This paper notes and explains some of the severe challenges posed by spatial interdependence in binary-outcome models and then follows recent spatial-econometric advances to suggest two simulation approaches for surmounting the analytically intractable and computationally intense estimation demands of these models, frequentist recursive-importance-sampling (RIS) or Bayesian MCMC. In brief, the complications arise because the endogenous spatial-lag implies the conditional independence that typically yields likelihoods for maximization that simply multiply N univariate distributions will not obtain. With interdependent observations, the likelihood is instead one N-variate joint distribution, and the one N-dimensional cumulative-normal in spatial probit is tremendously more intense to compute than the N cumulative standard-normal distributions of the common probit. After discussing Monte-Carlo comparisons of the performance of these alternative estimators, including our own, we show how to apply the same estimation-by-simulation methods to calculate estimated spatial effects of counterfactual shocks in terms of outcomes or probabilities of outcomes (with associated confidence/credibility regions) rather than in terms of parameter estimates or latent variables only as in prior spatial-probit applications.