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