Two spatial non-nested tests for weight structure in the spatial autoregressive model

The spatial J-test (SJ-test) and the spatial JA-test (SJA-test) are outlined to determine the appropriate spatial weight structure for the spatial autoregressive model against a single, or a set of, non-nested alternatives. These two non-nested tests are based on the maximum likelihood framework and can be applied to both the simple spatial autoregressive model and the general spatial autoregressive model. The Monte Carlo simulation results showed that the SJ-test and the SJA-test had clearly distinct performance in determining the appropriate spatial weight structure for a spatial autoregressive model against a single alternative and against two alternatives. The SJ-test performed better than the SJA-test in failing to reject a true null hypothesis when compared against a single alternative, whereas the SJA-test performed better than the SJ-test in failing to reject a true hypothesis when compared against two alternatives. Therefore, the SJ-test should be applied to determine the spatial weight structure against a single alternative, and the SJA-test should be applied to determine the spatial weight structure against a set of alternatives. Performance improved with a larger sample size and with a larger magnitude of spatial autocorrelation. The power of the tests depended on the degree of similarity between the alternatives and the underling true structure. The power of the tests was likely to be lower with two alternatives rather than a single alternative. This is due to the introduction of a false spatial weight matrix into the alternative with the true spatial weight matrix.