  The classic integrated conditional moment test is a promising method for
testing regression model misspecification. However, it severely suffers from
the curse of dimensionality. To extend it to handle the testing problem for
parametric multi-index models with diverging number of covariates, we
investigate three issues in inference in this paper. First, we study the
consistency and asymptotically linear representation of the least squares
estimator of the parameter matrix at faster rates of divergence than those in
the literature for nonlinear models. Second, we propose, via sufficient
dimension reduction techniques, an adaptive-to-model version of the integrated
conditional moment test. We study the asymptotic properties of the new test
under both the null and alternative hypothesis to examine its ability of
significance level maintenance and its sensitivity to the global and local
alternatives that are distinct from the null at the fastest possible rate in
hypothesis testing. Third, we derive the consistency of the bootstrap
approximation for the new test in the diverging dimension setting. The
numerical studies show that the new test can very much enhance the performance
of the original ICM test in high-dimensional scenarios. We also apply the test
to a real data set for illustrations.
