# Non-linear spatial models

## Non-linear spatial regressions

In Chapter 3 we considered the case of linear spatial regressions. In this chapter we will extend the analysis to non-linear models. Non-linear models can emerge for different reasons. One important instance is when the dependent variable can assume only a limited number of discrete outcomes. There are many examples of cases where this modeling framework is useful in spatial microeconometrics. For instance, we might be interested in explaining patient’s choices in health economics or school choices in educational economics. But many other examples can be found: for instance, when studying the presence or absence of a certain technology in a set of firms in industrial economics, consumer choice regarding different shopping centers, in electoral behavior, in criminal behavior and in a large number of other situations. We refer to these cases as to “discrete choice modeling” (see Greene, 2018). A second important class of non-linear models emerges when the dependent variable is limited by censoring or truncation (Greene, 2018). Surprisingly, despite the interest of non-linear models from an applied perspective, in the spatial econometrics literature they have received comparatively less attention than the models presented in Chapter 3, partly because of the higher analytical complexity involved and the associated higher computational effort required even in moderately large samples. In the case of non-linear models the framework presented in Chapter 3 cannot be employed, and it needs to be adjusted to respect the statistical nature of the datasets employed. The various specifications of spatial non-linear models follow the general strategy used in the literature to deal with non-spatial non-linear models with a particular emphasis on the “logit”, “probit” and “tobit” specifications adapted to account for the presence of spatial dependence in the dependent variable. In what follows we will present some of the most popular spatial versions of these models. The interested reader is referred to the works of Beron and Vijverberg (2004), Fleming (2004), Smirnov (2010) and LeSage and Pace (2009) for more thorough reviews.