Parametric and Non-parametric Encompassing Procedures

We study the asymptotic behaviour of encompassing statistics in general regression models. The theory for testing one parametric model against another parametric model is now well known, but the comparison of two non-parametric models, or 'crossed' Situations where a parametric model is tested against a non-parametric one, has not been treated previously. The encompassing test statistics for the four cases presented here are based on an appropriately normalized difference between an estimator of All, parameters (eventually functional), and its pseudo-true value under M(1). The specification tests for non-parametrically estimated models have meaning only when the smoothing parameter is not arbitrarily chosen, and so the window widths are calculated by ail automatic empirical method (cross-validation). As the window width determination is part of the estimation procedure, the pseudo-true window width, associated with the pseudo-true value, is defined.