A study on geographically weighted spatial autoregression models with spatial autoregressive disturbances

Abstract

Spatial heterogeneity and correlation are both considered in the geographical weighted spatial autoregressive model. At present, this kind of model has aroused the attention of some scholars. For the estimation of the model, the existing research is based on the assumption that the error terms are independent and identically distributed. In this article we use a computationally simple procedure for estimating the model with spatially autoregressive disturbance terms, both the estimates of constant coefficients and variable coefficients are obtained. Finally, we give the large sample properties of the estimators under some ordinary conditions. In addition, application study of the estimation methods involved will be further explored in a separate study.

Keywords:

• Mixed geographically weighted regression
• spatial autoregressive
• two stage least squares
• large sample properties

Acknowledgments

The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this article.

Notes

1 In the application, there are three kinds of choice of weight function as follow:

Gauss distance weight function: $Q_j(u_i,v_i)=\varphi \left(\frac{d_{\mathit{ij}}}{\sigma h}\right),$ Exponential distance weight function: $Q_j(u_i,v_i)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\frac{d_{\mathit{ij}}}{h}\right)}^2\right),$ Cubic distance weight function: $Q_j(u_i,v_i)={\left(1-{\left(\frac{d_{\mathit{ij}}}{q_i}\right)}^3\right)}^{3}I(d_{\mathit{ij}}<q_i),$ Where $d_{\mathit{ij}}$ is the distance between $(u_i,v_i)$ and $(u_j,v_j),$ $\varphi$ is the density function of standard normal distribution, $\sigma$ is the standard deviation of $d_{\mathit{ij}},$ $h$ is the window width, $q_i$ is the distance between the observed value $i$ to the nearest neighbor $q,$$I(\cdot )$ is the indication function, $j=1,2,\dots ,n.$

Additional information

Funding

This work was supported by National Social Science Foundation Project of China (No. 16BGL033).