Model checking for regressions: An approach bridging between local smoothing and global smoothing methods

For regression models, most of the existing model checking tests can be categorized into the broad class of local smoothing tests and of global smoothing tests. Compared with global smoothing tests, local smoothing tests can only detect local alternatives distinct from the null hypothesis at a much slower rate when the dimension of predictor vector is high, but can be more sensitive to oscillating alternatives. A projection-based test is suggested in multivariate scenarios to bridge between the local and global smoothing-based methodologies such that a local smoothing test can be transferred to a global smoothing test and still, to a certain extent, inherits some feature of local smoothing tests. The test construction rests on a kernel estimation-based method and the resulting test becomes a distance-based test with a closed form. Although it is eventually similar to an Integrated Conditional Moment test in spirit, it results in a test with a weight function that helps to collect more information from the samples than Integrated Conditional Moment test. Simulation results show that the proposed test has better performance than some typical competitors in this area when dimension goes higher. A real data example is analyzed to show its usefulness.