Spatial Error Model

Introduction

The spatial error model, as Anselin (1988: 35) points out combines “the linear regression model with a spatial autoregressive disturbance” as follows:

$$y = X\beta + \mu \\ \mu = \lambda W\mu + \epsilon \\ \epsilon \sim N(0,\sigma^2 I_{n})$$

where $$\mu$$ is the disturbance term, which has its own spatial autoregressive structure, where $$\lambda$$ is the coefficient of the spatially lagged error term. The presence of the spatially autocorrelated error term “is analogous to the serial correlation problem in time series models” (LeSage, 1998: 50). This model is traditionally used if it is determined that there is spatial autocorrelation in the residuals of a standard least squares regression.

Inputs

• Dataset – the spatial weight matrix to be used, probably derived from one of the methods above.
• Dataset – the dataset that contains the variable(s) to be tested.
• Spatial Regression Dependent Variable – the dependent variable(s) of the regression equation.
• Spatial Regression Independent Variables – the independent variable(s) of the regression equation.

Outputs

• The estimate for the regression coefficients, their standard error, Z value and p value
• The estimate for $$\rho$$, its standard error, Z value and p value
• Sigma squared ($$\sigma^2$$)
• Sigma ($$\sigma$$)
• Log Likelihood (LL)
• Akaike Information Criterion (AIC)
• Likelihood Ratio test result
• Wald test result
• The residuals
• Asymptotic coefficient covariance matrix for ($$\sigma$$, $$\rho$$ and $$\beta$$)
• The variable that was used
• Residual plots that the user requests