Cox-type tests for competing spatial autoregressive models with spatial autoregressive disturbances

• We derive the Cox-type tests of non-nested hypotheses for SARAR models.
• The Cox-type and J-type tests for SARAR models are not asymptotically equivalent.
• We show that the bootstrap is consistent for Cox-type tests.
• The Cox-type tests have relatively high power compared to other specification tests.

In this paper, we consider the Cox-type tests of non-nested hypotheses for spatial autoregressive (SAR) models with SAR disturbances. We formally derive the asymptotic distributions of the test statistics. In contrast to regression models, we show that the Cox-type and J-type tests for non-nested hypotheses in the framework of SAR models are not asymptotically equivalent under the null hypothesis. The Cox test in a non-spatial setting has been found often to have large size distortion, which can be removed by bootstrap. Cox-type tests for SAR models with SAR disturbances may also have a large size distortion. We show that the bootstrap is consistent for Cox-type tests in our framework. Performances of the Cox-type and J-type tests as well as their bootstrapped versions in finite samples are compared via a Monte Carlo study. These tests are of particular interest when there are competing models with different spatial weight matrices. Using bootstrapped p-values, the Cox tests have relatively high power in all experiments and can outperform J-type and several other related tests in some cases.