Testing Heteroscedasticity In Single-index Model

Semi-parametric regression model is a kind of important dimension reduction model.Because of this model can retain the advantage of non-parametric smoothing while achieving data dimension reduction,so it has strong interpretability and robustness.A important class of the semi-parametric regression model is the single-index model and the partial linear single-index model.A standard assumption is the independent and homogeneity of the error variance in regression models.Violation of this assumption will lead to a number of problems,such as the parameter estimate is not efficient;the hypothesis test is meaningless.Therefore,it is particularly important to test the heteroscedasticity of the model before statistical inference.Some scholars studied with heteroscedasticity in regression model,such as partial linear mode and partial linear varying-coefficient model.However,there are few studies on the problem of heteroscedasticity test for the single-index model and the partial linear single-index model.In this paper,the diagnostics for heteroscedasticity in the single-index model and partial linear single-index model are studied.Firstly,we estimate the unknown parameters and link function in test statistics by using kernel smoothing method,least squares method and estimate equation methods.Secondly,empirical likelihood method is used to construct empirical likelihood ratio test statistics.Then,under the null hypothesis,the theorem that the statistic is asymptotically distributed as chi-square distribution is obtained.Next,R software is used for the simulations,and the simulation results show that the test statistic has good properties in the inspection level and efficiency.At the same time of simulation study,we choose American cigarette consumption data for empirical analysis.The analysis results show that heteroscedasticity test method is effective and practical.Finally,the theorem presented in this paper is proved.