  This paper discusses two goodness-of-fit testing problems. The first problem
pertains to fitting an error distribution to an assumed nonlinear parametric
regression model, while the second pertains to fitting a parametric regression
model when the error distribution is unknown. For the first problem the paper
contains tests based on a certain martingale type transform of residual
empirical processes. The advantage of this transform is that the corresponding
tests are asymptotically distribution free. For the second problem the proposed
asymptotically distribution free tests are based on innovation martingale
transforms. A Monte Carlo study shows that the simulated level of the proposed
tests is close to the asymptotic level for moderate sample sizes.
