We establish a weak invariance principle for certain functionals of the regressogram estimator for regression or autoregression models where the data are strongly mixing. These functionals are constructed by cumulating the local discrepancies between the regressogram estimator and the corresponding regression function. As a byproduct, we obtain the limiting distribution of these functionals. Since the limiting process turns out to be a tractable time-changed Wiener process, we can derive from our results a family of possible nonparametric goodness-of-fit tests for the restriction to any compact interval of the regression or autoregression function. We then focus on a specially interesting test within this family. Using our preceding results, we provide estimates for the asymptotic behavior of the power of this test against both fixed and local alternatives.