Rao's score test in spatial econometrics

Rao's score test provides an extremely useful framework for developing diagnostics against hypotheses that reflect cross-sectional or spatial correlation in regression models, a major focus of attention in spatial econometrics. In this paper, a review and assessment is presented of the application of Rao's score test against three broad classes of spatial alternatives: spatial autoregressive and moving average processes, spatial error components and direct representation models. A brief review is presented of the various forms and distinctive characteristics of RS tests against spatial processes. New tests are developed against the alternatives of spatial error components and direct representation models. It is shown that these alternatives do not conform to standard regularity conditions for maximum likelihood estimation. In the case of spatial error components, the RS test does have the standard asymptotic properties, whereas Wald and Likelihood Ratio tests do not. Direct representation models yield a situation where the nuisance parameter is only identified under the alternative, such that a Davies-type approximation to the significance level of the RS test is necessary. The performance of both new RS tests is illustrated in a small number of Monte Carlo simulation experiments.