Study On Variable Selection In Expectile Regression And Optimal Portfolio Selection

High-dimensional data arise more and more frequently in economics, finance, s-tatistics, genetic engineering and related fields. When dealing with high-dimensional data, it is a fundamental task to identify the underlying model structure. In statis-tics, regression is one of the popular models to describe the data structure, such as ordinary least squares regression and quantile regression. With the development of statistical science and computing capability, variable selection methods in re-gression have been evolved from best subset regression or stepwise regression to regularized framework for multi-dimensional or high-dimensional data. Tibshirani[89] introduced L1 restriction on the regression coefficients in ordinary least squares regression, i.e. Lasso method to perform variable selection. Since then, Fan and Li[30]proposed the SCAD penalty to attain model consistency and asymptotic normal distribution, the so-called "oracle" properties while Lasso-type methods don't share.Regularized methods develop rapidly and play a more and more important role in variable selection during regression. In this work, we apply regularized framework to Expectile regression, proposed by Newey & Powell [70](1987)and perform the following two researches:· When the cardinality of the regression coefficients is fixed, we study variable selection and coefficient estimation in penalized Expectile regression and prove this method shares the "oracle" properties.· When dealing with high-dimensional data, we study variable selection in pe-nalized Expectile regression with the SCAD penalty under the situation where the regression error only has finite moments and point out the oracle estimator is one of the local solutions to the corresponding optimization problem. We adopt the CCCP algorithm to solve this nonconvex optimization problem and prove that the proposed CCCP algorithm can converge to the oracle estimator after several iterations. This result is desired in practice since it unifies the local optimum computed by this algorithm and the theoretical one. Besides,when heteroscedasticity exists, our results demonstrate this method can pick up those variables resulting in heteroscedasticity while the classical ordinary least squares regression fails.Besides, we study the optimal portfolio selection under data-driven non-parametric Mean-CVaR framework, especially, when short-selling is allowed, we introduce L1 re-striction on strategy allocation so that variable selection techniques can be applied to portfolio selection. We investigate the asymptotic behaviors of the optimal port-folio theoretically and prove that the proposed model can work well from the aspect of consistency in optimal solution and efficient frontier.