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

Abstract

We have argued and shown elsewhere the ubiquity and prominence of spatial interdependence, i.e., interdependence of outcomes among cross-sectional units, across the theories and substance of political and social science, and we have noted that much previous practice neglected this interdependence or treated it solely as nuisance, to the serious detriment of sound inference. These earlier studies considered only linear-regression models of spatial/spatio-temporal interdependence. For those classes of models, we (1) derived analytically in simple cases the biases of non-spatial and spatial least-squares (LS) under interdependence, (2) explored in simulations under richer, more realistic circumstances the properties of the biased non-spatial and spatial least-squares estimators and of the consistent and asymptotically efficient spatial method-of-moments (i.e., IV, 2SLS, GMM) and spatial maximum-likelihood estimators (ML), and (3) showed how to calculate, interpret, and present effectively the estimated spatial/spatio-temporal effects and dynamics of such models, along with appropriate standard errors, confidence intervals, hypothesis tests, etc. This paper begins a like set of tasks for binary-outcome models. We start again by stressing the ubiquity and centrality substantively and theoretically of interdependence in binary outcomes of interest to political and social scientists. We note that, again, this interdependence has typically been ignored in most contexts where it likely arises and that, in the few contexts where it has been acknowledged, or even rather centrally emphasized, those of policy diffusion and of social networks, the endogeneity of the spatial lag used (appropriately) to model the interdependence has only rarely been recognized. Next, we note and explain some of the severe challenges for empirical analysis posed by spatial interdependence in binary-outcome models, and then we follow recent advances in the spatial-econometric literature to suggest Bayesian or recursive-importance-sampling (RIS) approaches for tackling the estimation demands of these models. In brief and in general, the estimation complications arise because among the RHS variables is an endogenous weighted spatial-lag of the unobserved latent outcome, y*, in the other units; Bayesian or RIS techniques facilitate the complicated nested optimization exercise that follows from that fact. We show how to calculate estimated spatial effects (as opposed to parameter estimates) in such models and how to construct confidence regions for those, adopting simulation strategies for these purposes, and then how to present such estimates effectively.

Keywords: Spatial Econometrics, Spatial Interdependence, Spatial Lag, Spatial Probit, Gibbs Sampler, Metropolis-Hastings Sampler, Recursive Importance Sampler

JEL Classification: C1, C3, C5, C51, D31, D72, D78, E62, F02, F42, H11, H53, J65