In the context of regression analysis it is known that the residual cusum process may serve as a basis for the construction of various omnibus, smooth and directional goodness-of-fit tests. Since a deeper analysis requires the decomposition of the cusums into their principal components and this is difficult to obtain, we propose to replace this process by its innovation martingale. It turns out that the resulting tests are (asymptotically) distribution free under composite null models and may be readily performed. A simulation study is included which indicates that the distributional approximations already work for small to moderate sample sizes.
"Model checks for regression: an innovation process approach." Ann. Statist. 26 (5) 1916 - 1934, October 1998. https://doi.org/10.1214/aos/1024691363