In this work we consider the problem of making inference on the nonparametric component within a semiparametric regression model with differential regularization. The parametric inference methods so far introduced in the literature perform poorly, due to the variance misspecification induced by the penalization term. Nonparametric inference procedures may instead be excessively computationally demanding. We hereby propose two new parametric approaches, that are robust to the effect of the penalization, while retaining a reduced computational cost. 

The first method relies on an appropriate undersmoothing strategy, combined with a bootstrap approach. The second one leverages instead on the asymptotic properties of the scores of the model. The resulting tests have better control of Type-I error, with respect to the existing alternatives, and reduced computational cost. We apply the novel approaches to the study chlorophyll-a concentrations in the Mediterranean sea.