# Spatial Lag Model

## Contents

## 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