  A longstanding problem of existing empirical process-based tests for
regressions is that when the number of covariates is greater than one, they
either have no tractable limiting null distributions or are not omnibus. To
attack this problem, we in this paper propose a projection-based
adaptive-to-model approach. When the hypothetical model is parametric
single-index, the method can fully utilize the dimension reduction model
structure under the null hypothesis as if the covariate were one-dimensional
such that the martingale transformation-based test can be asymptotically
distribution-free. Further, the test can automatically adapt to the underlying
model structure such that the test can be omnibus and thus detect alternative
models distinct from the hypothetical model at the fastest possible rate in
hypothesis testing. The method is examined through simulation studied and is
illustrated by a real data analysis.
