One for All and All for One: Regression Checks With Many Regressors

We develop a novel approach to building 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 only one regressor whatever the number of regressors. This test can be viewed as a smooth version of the integrated conditional moment 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 not very sensitive to the smoothing parameter and performs well in multidimensional settings. We apply this test to a cross-country growth regression model.