One for all and all for one : Dimension reduction for regression checks

We develop a novel dimension-reduction approach to consistent checks of parametric regression models when many regressors are present. The principle is to replace the nonparametric alternative by a class of semiparametric alternatives, namely single-index models, that is rich enough to allow detection of any nonparametric alternative. We propose an omnibus test based on the kernel method that performs against a sequence of directional local nonparametric alternatives as if there was one regressor only, whatever the number of regressors. This test can also be viewed as an improved integrated conditional moment test. A simulation study reveals that our test performs better than several known lack-of-fit tests in multidimensional settings. Our test is little sensitive to the smoothing parameter choice. Moreover, qualitative information can be easily incorporated in the procedure to further improve its power.