In the multivariate nonparametric regression model Y = gt(t)+∑ the problem of testing linearity of the regression function g and homoscedasticity of the distribution of the error e is considered. For both problems a simple test is derived which is based on estimating the L² distance between the model space and the space induced by the hypothesis. The resulting statistics can be shown to be asymptotically normal, even under fixed alternatives. This extends and unifies recent results of Dette and Munk (1998a,b) to the multivariate case. A small simulation study on the finite sample behaviour of the proposed tests is reported and their properties are illustrated by analyzing a data example.