Integrated conditional moment test for partially linear single index models incorporating dimension-reduction

Abstract

Studying model checking problems for partially linear single-index models, we propose a variant of the integrated conditional moment test using a linear projection weighting function, which gains dimension reduction and makes the proposed method act as if there exists only one covariate even in the presence of multiple dimensional regressors. We derive asymptotic distributions of the proposed test; i.e., an integral of a centered Gaussian process under the null hypothesis and an integral of a non-centered one under Pitman local alternatives. We also suggest a consistent bootstrap procedure for calculating the critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.