Comparison of separable components in different samples

Imagine we have two different samples and are interested in doing semi- or nonparametric regression analysis in each of them, possibly on the same econometric model. In this article we consider the problem of testing whether a specific covariate has different impacts on the regression curve in these two samples. So we compare regression curves of different samples but being interested in specific differences instead of testing for equality of the whole regression function. Our procedure does not only allow for random designs and different sample sizes but also for different variance functions, different sets of regressors with different impact functions, etc. Actually, it is as general as the comparison of particular coefficients in different parametric regression models but now on the level of (nonparametric) functionals. As we use the marginal integration approach this method can be applied to any strong, weak or latent separable model to compare the (lower dimensional) separable components between the different samples. Thus, in the case of separable models our procedure includes the possibility of comparing the whole regression curves avoiding the curse of dimensionality that otherwise would render such a task impractical. In practice, resampling methods are necessary for applying our test to real data. However, it will be shown that for our approach bootstrap fails in practice and theory. Instead, we propose a subsampling procedure with automatic parameter choice. We give complete asymptotic theory, and its excellent performance is demonstrated by an extensive simulation study.