One for all and all for one: Regression checks with many regressors

We develop a novel 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 a smooth version of the integrated conditional moment (ICM) test of Bierens. For these reasons, we label our test the smooth ICM test. Moreover, qualitative information can be easily incorporated in the procedure to further improve its power. In an extensive simulation study, we provide evidence that our test is little sensitive to the smoothing parameter and performs better than several known lack-of-fit tests in multidimensional settings.