One for All and All for One : Regression Checks with Many Regressors”

We develop a novel approach to build consistent checks of parametric re-gression models when many regressors are present, based on a class of richenough semiparametric alternatives, namely single-index models. We proposean omnibus test based on the kernel method that performs against a sequenceof directional nonparametric alternatives as if there was one regressor only,whatever the number of regressors. This test can be viewed as a smooth ver-sion of the integrated conditional moment (ICM) test of Bierens. Qualitativeinformation can be easily incorporated in the procedure to enhance power.Our test is little sensitive to the smoothing parameter and performs betterthan several known lack-of-¯t tests in multidimensional settings.