Specification Testing of Regression Models with Mixed Discrete and Continuous Predictors

This article proposes a nonparametric projection-based adaptive-to-model specification test for regressions with discrete and continuous predictors. The test statistic is asymptotically normal under the null hypothesis and omnibus against alternative hypotheses. The test behaves like a locally smoothing test as if the number of continuous predictors was one and can detect the local alternative hypotheses distinct from the null hypothesis at the rate that can be achieved by existing locally smoothing tests for regressions with only one continuous predictor. Because of the model adaptation property, the test can fully use the model structure under the null hypothesis so that the dimensionality problem can be significantly alleviated. A discretization-expectation ordinary least squares estimation approach for partial central subspace in sufficient dimension reduction is developed as a by-product in the test construction. We suggest a residual-based wild bootstrap method to give an approximation by fully using the null model and thus closer to the limiting null distribution than existing bootstrap approximations. We conduct simulation studies to compare it with existing tests and two real data examples for illustration.