Summary: In regression analysis, it is important to test the validity of the assumed model prior to making inferences regarding the population of interest. In this investigation, we utilize nonparametric regression techniques to test the validity of a \(k\) th order polynomial regression model. The departures from the polynomial model are assumed to belong to a smooth class of functions; a parametric form is not assumed. A test based on nonparametric regression fits to the residuals from \(k\) th order polynomial regression is proposed. It utilizes a smoothing spline fit of order \(2k\) to the residuals from \(k\) th order polynomial regression. A test statistic based on this estimator is formulated and its asymptotic distribution is derived under alternatives converging to the null at a rate of \((n \lambda^{1/4 k})^{-1/2}\), where \(\lambda\) is the smoothing parameter. We note that this rate of convergence is slower than the parametric rate of \(n^{-1/2}\). Power investigations are conducted through a small-scale simulation study.