Specification testing of partially linear single-index models: a groupwise dimension reduction-based adaptive-to-model approach

This paper develops a groupwise dimension reduction-based adaptive-to-model test for partially linear single-index models. The test behaves as a local smoothing test would if the model were bivariate. The test statistic under the null hypothesis is asymptotically normally distributed. The test can detect local alternatives distinct from the null hypothesis at the rate that existing local smoothing tests can achieve when the regression model contains bivariate covariates. Therefore, the curse of dimensionality is largely alleviated. Numerical studies, including two real data examples, are conducted to examine the finite sample performance of the proposed test.