Asymptotic behavior of encompassing test for independent processes: Case of linear and nearest neighbor regressions

Encompassing test has been well developed for fully parametric modeling. In this study, we are interested on encompassing test for parametric and nonparametric regression methods. We consider linear regression for parametric modeling and nearest neighbor regression for nonparametric methods. We establish asymptotic normality of encompassing statistic associated to the encompassing hypotheses for the linear parametric method and the nonparametric nearest neighbor regression estimate. We also obtain convergence rate depending only on the number of neighbors k while it depends on the number of observation n and the bandwidth hₙ for kernel method. We achieve the same convergence rate when hₙ = k/n. Moreover, asymptotic variance of the encompassing statistic associated to kernel regression depends on the density, this is not the case for nearest neighbor regression estimate.