# 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