Variable Selection And Heteroscedasticity Identification Based On Quantile Regression

The ordinary least squares(OLS)is an important parameter estimation method in re-gression.It has good theoretical properties and conclusions when some conditions are satis-fied.However,OLS method has certain defects,such as the instability to outliers.Quantile regression(QR)method has focused mainly on estimating the quantile function of response.QR method is robust to outliers,and theoretical property does not need a strong assumption,so it has been deepening the methods such as parameter estimation,model identification,variable selection,etc.Furthermore,composite quantile regression(CQR)method has pro-posed to simultaneously consider multiple quantile regression models,it greatly improves the estimation efficiency.With the development of variable selection method,we combining QR method with variable selection method.For example,the lasso regularized quantile regression(LQR)method and the simultaneous multiple quantile regression(SMQR)method combind lasso penalty and sup-norm penalty respectively,they have made model selection by shrinking the fitted coefficients.However,these methods don't posse the oracle proper-ty.Adaptive lasso regularized quantile regression(ALQR)method and Adaptive sup-norm regularized simultaneous multiple quantiles regression(ASMQR)method have used adaptive weights for penalizing different coefficients in penalty,so that the estimated result has oracle property.Although these methods effectively achieve parameter estimation and model selection,the heteroscedasticity of the model is not discussed.Generally,QR method assumes that the model structure is specified,i.e.the varying and constants coefficients in the model are known.In this paper,we consider the parameter estimation and variable selection when the model structure is unknown.Furthermore,we will find the constants and varying coefficients in the model by the penalty method,solve the problem of unknown model structure,and discuss the heteroscedasticity of the model.This dissertation contains four chapters.Chapter one gives an overview of the research on QR method and variable selection method based on QR.Chapter two is the main content of this paper,including model introduction and main results.In this chapter,we propose LCQR method,which combines the variable selection method with adaptive group lasso penalty on the basis of CQR method,simultaneously estimates the coefficients of multiple quantile models,and retains the important variables in the model.Furthermore,LCQR method considers the relationship among coefficients in different quantile models,finds the varying and constants coefficients in the model by the penalty method of constant effect,and finds the model heteroscedasticity.At last,we prove that the estimation are consistent and spare.In the chapter three,we conduct numerical simulations and data analysis.We have conducted the numerical simulation under the homoscedastic and heteroscedastic linear model respectively,and obtain good results.In order to illustrate the effectiveness of the proposed method in practice,we uses LCQR method to analyzes Boston housing data,and shows that this method can realize variable selection and heteroscedasticity identification.Chapter four is the summary of this paper.