Abstract
Applications of spatial probit regression models that have appeared in the literature have incorrectly interpreted estimates from these models. Spatially dependent choices frequently arise in various modeling scenarios, including situations involving analysis of regional voting behavior, decisions by states or cities to change tax rates relative to neighboring jurisdictions, decisions by households to move or stay in a particular location. We use county-level voting results from the 2004 presidential election as an illustrative example of some issues that arise when drawing inferences from spatial probit model estimates. Although the voting example holds particular intuitive appeal that allows us to focus on interpretive issues, there are numerous other situations where these same considerations come into play. Past work regarding Bayesian Markov Chain Monte Carlo estimation of spatial probit models from LeSage and Pace (Introduction to spatial econometrics. Taylor and Francis, New York, 2009) is used, as well as derivations from LeSage et al. (J R Stat Soc Ser A Stat Soc 174(4):1007–1027, 2011) regarding proper interpretation of the partial derivative impacts from changes in the explanatory variables on the probability of voting for a candidate. As in the case of conventional probit models, the effects arising from changes in the explanatory variables depend in a nonlinear way on the levels of these variables. In non-spatial probit regressions, a common way to explore the nonlinearity in this relationship is to calculate “marginal effects” estimates using particular values of the explanatory variables (e.g., mean values or quintile intervals). The motivation for this practice is consideration of how the impact of changing explanatory variable values varies across the range of values encompassed by the sample data. Given the nonlinear nature of the normal cumulative density function transform on which the (non-spatial) probit model relies, we know that changes in explanatory variable values near the mean may have a very different impact on decision probabilities than changes in very low or high values. For spatial probit regression models, the effects or impacts from changes in the explanatory variables are more highly nonlinear. In addition, since spatial models rely on observations that each represent a location or region located on a map, the levels of the explanatory variables can be viewed as varying over space. We discuss important implications of this for proper interpretation of spatial probit regression models in the context of our election application.
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Notes
Specifically, the model that is implied by this type of omitted variable bias is referred to as the spatial Durbin probit model.
The term autoregressive refers to any spatial econometric model that includes a spatially lagged y variable (i.e., Wy) which includes the spatial autoregressive probit model, the spatial Durbin probit model, and their variants.
Of course, Albert and Chib (1993) did not deal with the case of spatial dependence, so \(\rho =0\) in their independent probit model.
Here again, the general definition of \(X \beta \) should be kept in mind, which would lead to a modification of \(\beta _v\) to \((\beta _v + W \theta _v)\) in expression (5) for the case of the SDM model.
For the case of the SDM model, \(\beta _v\) would be replaced by \((\beta _v + W \theta _v)\), as noted previously.
Again, our notation \(X \hat{\beta }= \sum _{v=1}^{k} (x_{v} \hat{\beta }_{v} +Wx_{v} \hat{\theta }_{v})\) is quite general and can also be used for the case of the SDM model.
Higher-order neighbors are neighbors to the neighbors, neighbors to these neighbors, and so on.
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Acknowledgements
James P. LeSage would like to acknowledge support for this research provided by the National Science Foundation, SES-0729264 and BCS-0136229. The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.
