Adjusted Rao's Score Test for Non-linear Hypotheses under Distributional and Local Parametric Misspecification

In this paper, we provide a general account to the asymptotic properties of a modified Rao's score (RS) statistic for testing a non-linear hypothesis under both distributional and parametric misspecification. The distributional misspecification arises if the parametric family of distribution functions that is used to formulate a quasi log-likelihood function does not include the data generating process (DGP). In the case of parametric misspecification, some of nuisance parameters, that are relevant for describing the DGP, are excluded from the model under the alternative hypotheses. We suggest a modified version of the score statistic that is robust to both types of misspecification. We establish the asymptotic properties of the robust test statistic under the null, the local alternatives, and the global alternative hypothesis. We provide four examples to illustrate the implementation of the robust score statistic. Our results demonstrate that the robust score statistic has good finite sample properties in terms of empirical size and power