Simulation And Empirical Study Of Heteroscedasticity Tests In Semi-parametric Regression Models

The semi-parametric regression model attracts lots of attention in statistical society be-cause of its beneficial properties as it combines the parametric and the nonparametric regression,and due to its wide applications in econometrics and other fields.As known that the consid-erable regression model in statistical society demands that this model has reliable parameters.Heteroscedasticity refers to the inequality of plotting the residuals of the model to the response variable.It is one of the most common violations for the assumptions of estimating the regres-sion model(in the ordinary least square,the most important assumption tells that the equal variances are set).Due to the importance of detecting Heteroscedasticity in the regression model for obtaining reliable parameters or applying statistical tests,this work provides a comprehen-sive work for checking the equality or inequality of variance based on the empirical properties of the tests and by using the simulation analysis.The main content of this dissertation is arranged as follows:Chapter 1 introduces a general description of the problem of this study.This chapter proposed the research background,significance,status,and existing problems for checking the heteroscedasticity in the regression models.Besides,we identify the objectives of the study and the reasons which lead to going deeply through this research.Moreover,it clarifies the outline of this dissertation about our given major work encompassing innovative keys.Chapter 2 discusses the literature review and the preparation information about applying the simulation analysis in order to detect heteroscedasticity in the regression models.The procedure step by step in this chapter mainly focuses on four topics.Firstly,introduce the simulation methods and how to apply its tools generally in statistical society and especially in regression models.Secondly,we discuss the semi-parametric regression model while it is conducted with two parts,the parametric and the nonparametric.We introduce the estimation methods for this model.Thirdly,we discuss the concept of heteroscedasticity and the most commonly used tests for it in regression models.Finally,we prepare some essential tools which are used to estimate and compare the test statistics of the heteroscedasticity in regression models such as Monte Carlo simulation(MC)and the analysis of variance(ANOVA).Chapter 3 is devoted to testing the heteroscedasticity in Semi-parametric models(the partially linear model in this chapter(PLM)).In this work,we established a new comprehensive test for heteroscedasticity comparing to the competitive methods for various situations.The test statistic is based on Levene's test and develops the null hypothesis and the alternative when the number of factors level goes to infinity.In this work,we consider that the errors are randomly generated from the standard normal distribution in the first case and t-distribution in the following situation.The main feature of our proposed method that it checks the influence of?X_ion?(X_i)directly with a lower impact of heavy-tailed distributions.Simulation studies are prepared for our test's power and have strong linear trends of the variance function.Also,simulations expected to show that the proposed test achieves better than existing methods,and the test is resilient to heavy-tailed error distributions even if the variance functions with strong nonlinear trends.Finally,the results of two Monte Carlo experiments are presented to examine the finite sample performances of the proposed procedures,and one empirical example is discussed.Chapter 4 prepares the test for heteroscedasticity in the single-index model(SIM),the mod-el which is commonly considered for high-dimensional regression analysis.This model specifies that it is more flexible compared to a parametric model,and avoids the curse of dimensionality.Single index reduces the dimensionality of a standard variable vector(X in multiple regression)to a univariate index(?X in the single index model).In this chapter,we develop a unique index regression model with a functional error term that serves to verify heteroscedasticity.S-ince the efficient interpreting of a regression model requires that heteroscedasticity be considered when it exists,this research presents the assumptions of testing the constancy of variance in single-index models.The test statistic is evaluating the homogeneity of variance established as the combination of the Levene test and the recent ANOVA theories for infinite factor levels.The test statistic in simulation studies is adequately displayed in all situations compared to a known method and is applied to a real case study.Chapter 5 proposes test statistics for partially linear single-index models based on the paired distances of the sample points to test heteroscedasticity.The test statistic formulated as U does not need to estimate the function of conditional variance through the use of nonpara-metric methods,such as the kernel,the local polynomial,and the spline.In this work,under the null hypothesis,we derive a computationally feasible approach to deal with the complexity of the zero limit distribution.Besides,we demonstrate that the bootstrap procedure is a valid approximation to the null distribution of the proposed test.This test statistic has an asymptot-ically normal distribution with a non-zero mean and an identical asymptotic variance.The test feature that the test method of the algorithmic program is easy to implement and has a faster convergence than some existing methods.The convergence rate of the test does not depend on the dimensions of the covariates,which significantly reduces the impact of the dimensional curse.Finally,we propose the numerical simulations and the application of the test through an example of real data.