Generalized Expectile Regression And Its Application In Financial Risk

The financial market plays an essential role in the development and opening up of economy.Once financial market risk rise,economy will be destroyed badly,and maybe break out economic crisis,and maybe bring social turbulence.Therefo-re,it is important to guard against financial venture and maintain the national economic security.Meanwhile,this is the major issue we need to solve.The measure of risk is the key of risk management and control.In many of the risk metrics model,Value-at-risk(VaR)approach has been considered to be the standard measure of international financial risk.There are a variety of different viewpoints about how to compute VaR.The core of the methods are to estimate the statistical distribution or probability density function of the earnings of financial position.This paper attempts to use a new quantile regression method which is different from the traditional methods to measure the risk.It is well known that quantile regression methods are robust because of their check functions of least absolute deviation.Owing to squared check functions,expectiles are more sensitive to the tails of distributions and more effective for the normal case than quantiles.Efron(1991)pointed out that the power loss function with k=1.5 is appealing as a compromise between the robustness of k=1 and the high normal theory efficiency of k=2.In this paper,we consider the kth power expectiles and the kth power expectile regression methods,mainly for the cases of 1<k<2 which can be seen as an important special one of M estimation methods In order to trade off these two aspects,we use a loss function which falls in between the above two loss functions and develop an asymmetric least kth power estimation method,i.e.,the kth power expectile regression.This work is a partial extension of those of both Koenker and Bassett(1978)and Newey and Powell(1987).We attempt to construct a bridge between quantiles and expectiles.The kth power expectile regressions are only based on the condition that the true distribution of the observations has(k-1)th order moment.When k=1 and k=2,these loss functions are ones used for the quantile regression and the expectile regression proposed by Koenker and Bassett(1978)and Newey and Powell(1987),respectively.Returning to k=1 and k=2,the consistency and asymptotic normality of estimator had been proved by Koenker and Bassett(1978)and Newey and Powell(1987),respectively.Since then,thanks to their great advantages,the quantile regression and expectile regression methods have been used in all over the fields of science.For a detailed and systematic introduction to quantile regression and some interesting extensions of basic quantilebased models,we refer to Koenker(2005),Engle and Manganelli(2004),Kim(2007),Cai and Xu(2008),Cai and Xiao(2012),Andriyana et al.(2016)and Koenker(2017),amongst others.More expectile-based models can be found in Efron(1991),Yao and Tong(1996),Granger and Sin(1997),Taylor(2008),Kuan et al.(2009),Gu and Zou(2016),Farooq and Steinwart(2017).For k=1,Koenker and Bassett(1978)considered completely the properties related to the regression quantiles.For proving the uniqueness of the minimum point they used the liner programming formulation and the theory of polyhedra.The algorithm of the estimator also comes from the liner programming.The asymptotic properties of the estimators were proved by the basic event probability and Scheff'e's theorem(Scheff'e(1947)).When 1<k<2,the object function is not of the liner and polyhedra,we mainly use the spirit of the methods of Huber(1967)and Newey and Powell(1987)to prove our theorems.Because the expressions related to kth power expectiles(1<k<2)are more complex,more mathematical techniques are used in the proofs such as in proving the existence-uniqueness of kth power expectiles and in proving Theorem 3.3.Some comparisons with the quantile regression and the expectile regression have been made in detail,which illustrates the advantage of the general kth power expectile regression.A practical example of analyzing the data of incomes of migrant workers is provided and exemplifies regression methods although our methods are applicable in a much broader context.It is worthy mentioning that the empirical results show that the 1.5th expectile regression can excavate much more information than the 2th expectiles regression,i.e.,the common expectiles regression.The paper consists of four chapter,the structure and contents are described as follows.The first chapter is the introduction of this paper.It discusses the background,ideas,significance and status of the research.The innovation of this research are introduced.The second chapter gives a brief historical overview of financial market risks and VaR which is a risk measurement tool.According to the evolution process of the measurement of financial market risks,this chapter introduces various methods of financial market measurement.The third chapter studies a new quantile regression model which loss function's power is k(1<k<2).The existence and uniqueness of kth power expectiles under mild conditions have been proved.Furthermore,we prove consistency and asymptotic normality of the estimators for the kth power expectile regression.Some comparison of kth power expectile estimators with estimators of quantile and expectile indicates the advantage of kth power expectile regression method.The property of kth power expectile regression is close to that of quantile regression as k approaches 1,while the property of kth power expectile regression gets close to that of expectile regression as k tends to 2.According to the specific problem and their preference and noting the results in subsection 3.2,researchers can choose the suitable k to run a satisfying kth power expectile regression.Simulation studies are conducted to examine the performance of the proposed methods.Results show that the efficiency of kth power expectile regression is higher than those of the common quantile regression and expectile regression in most cases when the underlying distribution is the t(3)distribution.In the real data analysis,we fit the Mincerian earnings function model to the data of incomes of migrant workers in China at 2011.Results show that 1.5 power expectile regression can excavate much more information than 2 power expectile regression,i.e.,common expectile regression.Thus the empirical analysis by using 1.5 power expectiles regression may bring more empirical evidence for the well-known theory of labor economics.The fourth chapter defines EVaR based on kth power expectile according to kth power expectile and kth power expectile regression which are studied in the third chapter.In order to apply asymmetric least square method advanced by Newey and Powell(1987)to dynamic state,we further extend asymptotic theory from the i.i.d data to stationary and weakly dependent data.These methods are similar to those in Chung-Ming Kuan,Jin-Huei Yeh?Yu-Chin Hsu(2009)and are deferred to CARE model and encompassing test.We conduct a empirical study to assess the value at risk of two stock indices,NASDAQ and Nikkei.Overall,the thesis sums up the advantages and the defects of the new expectile method and points out the future research work by summarizing the research content of the whole dissertation.