  There exist a number of tests for assessing the nonparametric heteroscedastic
location-scale assumption. Here we consider a goodness-of-fit test for the more
general hypothesis of the validity of this model under a parametric functional
transformation on the response variable. Specifically we consider testing for
independence between the regressors and the errors in a model where the
transformed response is just a location/scale shift of the error. Our criteria
use the familiar factorization property of the joint characteristic function of
the covariates under independence. The difficulty is that the errors are
unobserved and hence one needs to employ properly estimated residuals in their
place. We study the limit distribution of the test statistics under the null
hypothesis as well as under alternatives, and also suggest a resampling
procedure in order to approximate the critical values of the tests. This
resampling is subsequently employed in a series of Monte Carlo experiments that
illustrate the finite-sample properties of the new test. We also investigate
the performance of related test statistics for normality and symmetry of
errors, and apply our methods on real data sets.
