We consider an additive model with second order interaction terms. It is shown how the components of this model can be estimated using marginal integration, and the asymptotic distribution of the estimators is derived. Moreover, two test statistics for testing the presence of interactions are proposed. Asymptotics for the test functions are obtained, but in this case the asymptotics produce inaccurate results unless the number of observations is very large. For small or moderate sample sizes a bootstrap procedure is suggested and is shown to work well on a simulated example. Finally, our methods are illustrated on a five-dimensional production function for a set of Wisconsin farm data. In particular, the separability hypothesis for the production function is discussed.