  This paper studies model checking for general parametric regression models
having no dimension reduction structures on the predictor vector. Using any
U-statistic type test as an initial test, this paper combines the
sample-splitting and conditional studentization approaches to construct a
COnditionally Studentized Test (COST). Whether the initial test is global or
local smoothing-based; the dimension of the predictor vector and the number of
parameters are fixed or diverge at a certain rate, the proposed test always has
a normal weak limit under the null hypothesis. When the dimension of the
predictor vector diverges to infinity at faster rate than the number of
parameters, even the sample size, these results are still available under some
conditions. This shows the potential of our method to handle higher dimensional
problems. Further, the test can detect the local alternatives distinct from the
null hypothesis at the fastest possible rate of convergence in hypothesis
testing. We also discuss the optimal sample splitting in power performance. The
numerical studies offer information on its merits and limitations in finite
sample cases including the setting where the dimension of predictor vector
equals the sample size. As a generic methodology, it could be applied to other
testing problems.
