Partially Specified Spatial Autoregressive Model with Artificial Neural Network

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.