# First Order Spatial Lag

First Order Spatial Lag introduces spatial effects into a regression model, by including just a spatially lagged dependent variable:

$\LARGE{y = \rho Wy + \epsilon}$

$\LARGE{\epsilon \sim N(0,\sigma^2 I_{n})}$

Where $y$ is the n x 1 dependent variable vector (n being sample size), $W$ is the n x n spatial weight matrix, $\rho$ is the coefficient of the spatially lagged dependent variable, $W_{y}$ and $\epsilon$ the error term which is normally distributed with error variance matrix $\sigma^2 I_{n}$.

### SET UP

To demonstrate this tool, we will explore nodal connectivity of public transport in Melbourne’s West.

Select Melbourne – West SA4 (Australia → Victoria → Greater Melbourne → Melbourne – West) as your area.

Select SNAMUTS – Indicators by Areas (SA1) 2016 as your dataset, select Nodal Connectivity – Max Value as the variable.

Use the Spatialise Aggregated Dataset tool to Spatialise the dataset.

Use the Contiguous Spatial Weight Matrix tool to build a Spatial Weights Matrix for the spatialised dataset, using 1st order, row-standardised, Queen contiguity.

### Inputs

Open the First Order Spatial Lag tool (Tools → Spatial Regression → First Order Spatial Lag) and enter the following parameters:

• Dataset Input: The dataset that contains the variable to be tested. Select the Spatialised Dataset.
• Spatial Weights Matrix: The spatial weight matrix to be used. Select the Contiguous Spatial Weight Matrix.
• Key Column: Specify the unique codes for your areas. Select SA1 Main Code.
• Dependent Variable: The variable to be tested. Select Nodal Connectivity – Max Value.

The input parameters are summarised in the image below, once complete click Run Tool.

### Outputs

Once the tool has finished, tick both boxes and click Display Output. This will open up two outputs.

The first output is a data file that you can map, containing your input variables and the following:

• Residuals: the residuals (difference between observed value and the fitted value) for each dependent variable in the regression model.

The second output is a text window with a matrix of estimated parameters in the model:

• Rho:
• Estimate: The estimate for $\rho$.
• s.e.: The $\rho$ estimate standard error.
• z value: The $\rho$ estimate z-value.
• Pr(>|z|): The $\rho$ estimate p-value.
• Regression Coefficients:
• Estimate: The estimate for the regression coefficients.
• Std. Error: The regression coefficient standard error.
• z value: The regression coefficient z-value.
• Pr(>|z|): The regression coefficient p-value.
• Log Likelihood: The Log Likelihood (LL) value at computed optimum.
• Sigma squared: The residual variance ($\sigma^2$).
• Coefficient Covariance Matrix: Asymptotic coefficient covariance matrix for $\sigma$, $\rho$ and the intercept.
• Sigma: The Standard Deviation ($\sigma$).
• Akaike Information Criterion (AIC): The Akaike Information Criterion (AIC) of the linear model.
• Likelihood Ratio: Likelihood Ratio (LR) test result.
• Wald Test: Wald Test result.
• LM test on spatial lag residuals: Lagrange Multipliers test results performed on the residuals of the first order spatial lag regression.

### Looking for Spatial Data?

You can browse the AURIN Data Discovery: