Spatial Lag Model

Introduction

Also called the mixed regressive-spatial autoregressive model, we alter the first-order spatial autoregressive model by introducing a matrix of independent variables. This model combines the matrix of explanatory variables with a vector of coefficient parameters, with the spatially lagged dependent variable from the previous model:

\(y = \rho Wy + X\beta + \epsilon \\ \epsilon \sim N(0,\sigma^2 I_{n})\)

In this model, the spatially lagged dependent variable is correlated with the error terms (see Anselin, 1988: 58).

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
  • Lagrange Multiplier test performed on the residuals of the first order spatial lag regression
  • The residuals
  • Asymptotic coefficient covariance matrix for (\(\sigma\), \(\rho\) and \(\beta\))
  • The variable that was used
  • Residual plots that the user requests