Topology, Dependency Tests and Estimation Bias in Network Autoregressive Models

Regression analyses based on spatial datasets often display spatial autocorrelation in the substantive part of the model, or residual pattern in the disturbances. A researcher conducting investigations of a spatial dataset must be able to identify whether this is the case, and if so, what model specification is more appropriate for the data and problem at hand. If autocorrelation is embedded in the dependent variable, the following spatial autoregressive (SAR) model with a spatial lag can be used: 1$$\begin{array}{rcl} & & \mathbf{y} = \rho \mathbf{Wy} + \mathbf{X}\beta + \varepsilon , \\ & & \varepsilon \sim N(0,{\sigma }^{2}). \end{array}$$ On the other hand, when there is residual pattern in the error component of the traditional regression model, the spatial error model (SEM) can be used: 2$$\begin{array}{rcl} & & \mathbf{y} = \mathbf{X}\beta + \mathbf{u}, \\ & & \mathbf{u} = \rho \mathbf{Wu} + \varepsilon , \\ & & \varepsilon \sim N(0,{\sigma }^{2}).\end{array}$$ In the above equations, W is the spatial weight matrix representing the structure of the spatial relationships between observations, ρ is the spatial dependence parameter, u is a vector of autocorrelated disturbances, and all other terms are the elements commonly found in ordinary linear regression analysis.