This paper develops more accurate tests for lack of spatial correlation than ones based on the usual central limit theorem. We test nullity of the lag parameter in a pure spatial autoregression based on least squares and Gaussian maximum likelihood estimates. In each case, depending on assumptions on the spatial weight matrix, the rate of convergence of the estimate can be slower than the square root of n, where n is sample size. Correspondingly, the error in the normal approximation can be larger than the usual parametric order. This provides particularly strong motivation for employing instead refined statistics which entail closer approximations. These are based on (formal) Edgeworth expansions. In Monte Carlo simulations we demonstrate that the new tests (and one based on a bootstrap, which is expected to have similar properties) outperform one based on the usual normal approximation in small and moderate samples. The new tests are also applied in two empirical examples.

Lagrange multiplier tests of spatial uncorrelatedness in a pure spatial autoregressive model have advantages over other forms of testing. They are typically based on the chi square first-order asymptotic approximation to the distribution of the test statistic. In small samples this approximation may be poor. We develop refined tests based on Edgeworth expansion. These are compared by Monte Carlo simulations to ones that are respectively based on a bootstrap, and on the exact finite sample distribution. Generally such tests are found to significantly outperform those based on the chi square approximation. We also develop Edgeworth-based tests for uncorrelatedness of disturbances in a regression model, against the alternative of spatial autoregressive disturbances.

In this paper we focus on testing for spatial independence when the data follow a mixed spatial autoregressive model, i.e. a spatial autoregression where also a set of exogenous regressors is present. We test the nullity of the spatial parameter based on least squares and instrumental variables estimates. Since in much empirical work only small or moderately-sized samples are available, the error in the normal approximation may be large and hence testing conclusions unreliable. We derive higher order corrections for standard tests, which are expected to improve upon the approximation based on the central limit theorem. Such corrections are based on feasible formal Edgeworth expansions and bootstrap procedures. A Monte Carlo study confirms that the new tests outperform ones based on the central limit theorem in small and moderately-sized samples.