On the bootstrap for Moran’s I test for spatial dependence

This paper is concerned with the use of the bootstrap for statistics in spatial econometric models, with a focus on the test statistic for Moran’s I test for spatial dependence. We show that, for many statistics in spatial econometric models, the bootstrap can be studied based on linear–quadratic (LQ) forms of disturbances. By proving the uniform convergence of the cumulative distribution function for LQ forms to that of a normal distribution, we show that the bootstrap is generally consistent for test statistics that can be approximated by LQ forms, including Moran’s I. Possible asymptotic refinements of the bootstrap are most commonly studied using Edgeworth expansions. For spatial econometric models, we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of LQ forms. When the disturbances are normal, we prove the existence of the usual Edgeworth expansions for LQ forms; when the disturbances are not normal, we establish an asymptotic expansion of LQ forms based on martingales. These results are applied to show the second order correctness of the bootstrap for Moran’s I test.