An Alternative Asymptotic Analysis of Residual-Based Statistics

This paper offers an alternative technique to derive the limiting distribution of residual-based statistics or, more general, the limiting distribution of statistics with estimated nuisance parameters. This technique allows us to unify many known results on two-stage estimators and tests and we also derive new results. The technique is especially useful in situations where smoothness of the statistic of interest with respect to the parameters to be estimated does not hold or is difficult to establish, e.g., rank-based statistics. We essentially replace this differentiability condition with a distributional invariance property that is often satisfied in specification tests. Our results on statistics that have not been considered before all use nonparametric statistics. On the technical side, we provide a novel approach to the pre-estimation problem using Le Cam's third lemma. The resulting formula for the correction in the limiting variance as a result of pre-estimation some parameters is a simple expression involving some appropriate covariances. The regularity conditions required fairly minimal. Numerous examples show the strength and wide applicability of our approach