The Test For Heteroscedasticity In Single-Index Model

In the regression analysis, it is common to assume the independent homoscedasticity on the error term. If the assumption is violated, it will have great impact on the statistical inference. Hereby, the step to test the homoscedasticity of the error become the key step before setting up the model. In parametric and nonparametric regression models, the problem of testing heteroscedasticity was discussed widely. In this paper, the test of the heteroscedasticity , autocorrelation and the omission of an important variable in the single-index model were researched with the method of score test.Starting with the base ideology, in chapter 2, score tests are derived for three potential misspecifications of the single index model: heteroscedasticity in the errors, autocorrelation in the errors, and the omission of an important variable in the linear index. These tests have a similar structure to corresponding tests for nonlinear regression models. Simulations demonstrate that the first two tests hold their nominal size well, and have good power properties in identifying model violations, often outperforming other tests. Testing for the need for additional covariates can be effective, but is more difficult. The score tests are applied to two real data sets, demonstrating that the tests can identify important model violations that affect inference, and that approaches that do not take model misspecifications into account can lead to very different results.In final, we establish the asymptotic properties of score test statistics for the heteroscedasticity and autocorrelation of the score test, which is illustrated by the corresponding simulation.