Additive models are one means of assuaging the curse of dimensionality when nonparametric smoothing methods are used to estimate multivariable regression functions. It is important to have methods for testing the fit of such models, especially in high dimensions where visual assessment of fit becomes difficult. New tests of additivity are proposed in this paper that derive from Fourier series estimators with data-driven smoothing parameters. Other tests related to the classical Tukey test for additivity are also considered. While the new tests are consistent against essentially any "smooth" alternative to additivity, the Tukey-type tests are found to be inconsistent in certain situations. Asymptotic power of both varieties of tests is studied under local alternatives that tend toward additivity at a parametric rate, and small-sample power comparisons are carried out by means of a simulation study.
"Testing for additivity in nonparametric regression." Ann. Statist. 23 (6) 1896 - 1920, December 1995. https://doi.org/10.1214/aos/1034713639