  Spatial autoregressive model, introduced by Clif and Ord in 1970s has been
widely applied in many areas of science and econometrics such as regional
economics, public finance, political sciences, agricultural economics,
environmental studies and transportation analyses. As information technology
grows rapidly, observations are seldom independent from others so a space
autoregressive models can take this dependence into account and draw more
reliable conclusions between covariates and the target variable itself. Based
on the classical spatial model, Su and Jin proposed a semi-parametric model
named as partially specified spatial autoregressive model (PSAR) to allow for
more flexibility in modeling. And to estimate this nonparametric component, we
use the neural network model which adds more flexibility to the classical model
and allows for variations in the choice of activation functions according to
different types of data. This paper extends an artificial neural network model
to a partially specified space autoregressive model and proposes maximum
likelihood estimators instead of quasi-maximum likelihood estimates. We
establish the consistency and asymptotic normality of the estimators in this
model. These results are obtained under some standard conditions in spatial
models as well as neural network models. To illustrate, we investigate the
quality of the normal approximation for finite samples by means of numerical
simulation studies with three common choices of error distributions (standard
normal, student-t distribution and the Laplace distribution). We apply our
proposed model to a soil-water tension problem and a criminal study in Chicago.
The results showed that our model can capture the spatial dependence between
units as well as the unknown correlation structure between the target variable
and covariates.
