Consistent bootstrap tests of parametric regression functions

This paper introduces specification tests of parametric mean-regression models. The null hypothesis of interest is that the parametric regression function is correctly specified. The proposed tests are generalizations of the Kolmogorov–Smirnov and Cramer–von Mises tests to the regression framework. They are consistent against all alternatives to the null hypothesis, powerful against 1/n local alternatives, not dependent on any smoothing parameters and simple to compute. A wild-bootstrap procedure is suggested to obtain critical values for the tests and is justified asymptotically. A small-scale Monte Carlo experiment shows that our tests (especially Cramer–von Mises test) have outstanding small sample performance compared to some of the existing tests.