Heteroscedasticity testing for semi-parametric multi-index models based on the partial sufficient dimension reduction method

Abstract

Heteroscedasticity testing is an important issue in regression analysis. In this paper, a model-adaptive test statistic is proposed for the semi-parametric multi-index model. The semi-parametric model studied in this paper includes two types of predictors, namely the predictors of primary interest $X$ and the predictors of secondary interest $W$. In general, the dimension of the variable $X$ is relatively high, leading to the greatly reduced accuracy of the non-parametric estimation.Meanwhile, the existence of the secondary variable $W$ makes the traditional sufficient dimension reduction methods no longer applicable.In this paper, a dimension-reduction test statistic is constructed based on the partial sufficient dimension reduction method, which effectively avoids the estimation difficulties caused by high-dimensional variables. The statistic can be adaptive to many different model structures and thus has good robustness. The asymptotic properties of the proposed test statistic are studied under the null hypothesis and the alternative hypothesis. Simulations and real data examples are given to illustrate the finite sample performance of the test.