How do you choose the appropriate spatial weight matrix for spatial lagrange multiplier tests?

2 How to choose the appropriate spatial weights matrix?

There is no definitive answer to how to choose the appropriate spatial weights matrix, as different matrices may imply different assumptions and hypotheses about the spatial process. However, some general guidelines and criteria can help you make an informed decision. First, you should consider the spatial scale and resolution of your data and choose a matrix that reflects the relevant spatial interactions. For example, if you are analyzing county-level data, you may want to use a contiguity matrix that accounts for neighboring effects, while if you are analyzing point-level data, you may want to use a distance-based matrix that captures the spatial decay of influence. Second, you should test the sensitivity and robustness of your results to different spatial weights matrices and compare their fit and performance. For example, you can use the Lagrange Multiplier diagnostics to test for the presence and type of spatial dependence under different matrices, or use information criteria such as AIC or BIC to compare the goodness-of-fit of different models.

3 How to perform spatial lagrange multiplier tests?

Spatial lagrange multiplier tests are based on the comparison of the restricted and unrestricted models. The restricted model is the original regression model without any spatial terms, while the unrestricted model is the alternative model that includes a spatial term. The spatial term can be either a spatially lagged dependent variable, which implies a spatial autoregressive (SAR) process, or a spatially lagged error term, which implies a spatial error (SEM) process. The spatial lagrange multiplier test statistic is computed as the difference between the log-likelihoods of the restricted and unrestricted models, multiplied by two. The test statistic follows a chi-square distribution with one degree of freedom under the null hypothesis of no spatial dependence. If the test statistic is larger than the critical value, you can reject the null hypothesis and conclude that there is evidence of spatial dependence.

4 How to interpret the results of spatial lagrange multiplier tests?

The results of spatial lagrange multiplier tests can help you identify the type and direction of spatial dependence in your model. There are four main types of tests: LM-Lag, LM-Error, Robust LM-Lag, and Robust LM-Error. The LM-Lag and LM-Error tests are based on the assumption that there is only one type of spatial dependence in the model, while the Robust LM-Lag and Robust LM-Error tests are based on the assumption that there may be both types of spatial dependence in the model. The LM-Lag and Robust LM-Lag tests indicate whether there is a positive or negative spatial autocorrelation in the dependent variable, while the LM-Error and Robust LM-Error tests indicate whether there is a positive or negative spatial autocorrelation in the error term. If both types of tests are significant, you should use the Robust tests to determine which type of spatial dependence is more dominant.

5 How to correct for spatial dependence?

If you find evidence of spatial dependence in your model, you should correct for it by including the appropriate spatial term in your model. Depending on the type and direction of spatial dependence, you can use either a SAR model or a SEM model. A SAR model includes a spatially lagged dependent variable as an additional explanatory variable, while a SEM model includes a spatially lagged error term as an additional disturbance term. Both models require the specification of a spatial weights matrix, which should be consistent with the one used for the spatial lagrange multiplier tests. You can estimate these models using maximum likelihood or generalized method of moments techniques, and test for their significance and fit using likelihood ratio tests or information criteria.