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

We develop a novel approach to build checks of parametric regression models when many regressors are present, based on a class of sufficiently rich semiparametric alternatives, namely single-index models. We propose an omnibus test based on the kernel method that performs against a sequence of directional nonparametric alternatives as if there was one regressor only, whatever the number of regressors. This test can be viewed as a smooth version of the integrated conditional moment (ICM) test of Bierens. Qualitative information can be easily incorporated into the procedure to enhance power. In an extensive comparative simulation study, we find that our test is little sensitive to the smoothing parameter and performs well in multidimensional settings. We then apply it to a cross-country growth regression model.