Problem 1

SAR model with SAR term

SAR model without SAR term

From the residual plots, we can see that the residual from SAR model with the SAR term is more random and independent. It also has smaller error variance (0.035384) comparing to the other one (0.2893). It has a significantly smaller log likelihood (-132.0012) comparing to the OLS model (189.6171).

Problem 2

lagsarlm() is used to fit spatial simultaneous autoregressive lag model, where there is a lag term on the response variable. errorsarlm() is used to fit spatial simultaneous autoregressive error model estimation where the lag is on the error term.

Problem 3

There is no significant spatial dependence in the residual of the SAR model.

Problem 4

The two matrix are not equal, so the default is not symmetric. I’m not suprised. Two matrix is used, one uses a basic binary coding for neighbours, and the other uses a variance stabilizing coding scheme.

The first one is better in terms of log-likelihood and error variance.

Problem 5

Other than \(\beta_3\), the estimates are significantly biased. So the CAR model is more appropriate.