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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.

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