Bayesian expectile regression with asymmetric normal distribution

In this paper, we adopt the Bayesian approach to expectile regression employing a likelihood function that is based on an asymmetric normal distribution. We demonstrate that improper uniform priors for the unknown model parameters yield a proper joint posterior. Three simulated data sets were generated to evaluate the proposed method which show that Bayesian expectile regression performs well and has different characteristics comparing with Bayesian quantile regression. We also apply this approach into two real data analysis.     

Nonparametric multiple expectile regression via ER-Boost

Expectile regression [Newey W, Powell J. Asymmetric least squares estimation and testing, Econometrica. 1987;55:819–847] is a nice tool for estimating the conditional expectiles of a response variable given a set of covariates. Expectile regression at 50% level is the classical conditional mean regression. In many real applications having multiple expectiles at different levels provides a more complete picture of the conditional distribution of the response variable. Multiple linear expectile regression model has been well studied [Newey W, Powell J. Asymmetric least squares estimation and testing, Econometrica. 1987;55:819–847; Efron B. Regression percentiles using asymmetric squared error loss, Stat Sin. 1991;1(93):125.], but it can be too restrictive for many real applications. In this paper, we derive a regression tree-based gradient boosting estimator for nonparametric multiple expectile regression. The new estimator, referred to as ER-Boost, is implemented in an R package erboost publicly available at http://cran.r-project.org/web/packages/erboost/index.html. We use two homoscedastic/heteroscedastic random-function-generator models in simulation to show the high predictive accuracy of ER-Boost. As an application, we apply ER-Boost to analyse North Carolina County crime data. From the nonparametric expectile regression analysis of this dataset, we draw several interesting conclusions that are consistent with the previous study using the economic model of crime. This real data example also provides a good demonstration of some nice features of ER-Boost, such as its ability to handle different types of covariates and its model interpretation tools.     

基于ARCH-Expectile方法的VaR和ES尾部风险测量

甄别和确定风险因素的贡献是资产或资产组合风险管理的重要研究内容。近十年,下端风险越来越受到关注,在险价值（Value at Risk,VaR）和预期不足（Expected Shortfall,ES）是资产组合风险管理中两个常用的风险度量工具。Kuan等^[1]在一类条件自回归模型（CARE）下提出了基于expectile的VaR度量-EVaR。本文扩展了Kuan等^[2]的CARE模型到带有异方差的数据,引入ARCH效应提出了一个线性ARCH-Expectile模型,旨在确定资产或资产组合的风险来源以及评估各风险因素的贡献大小,并应用expectile间接评估VaR和ES风险大小。同时给出了参数的两步估计算法,并建立了参数估计的大样本理论。最后,将本文所提出的方法应用于民生银行股票损益的风险分析,从公司基本面、市场流动性和宏观层面三个方面选取影响股票损益的风险因素,分析结果表明,各风险因素随股票极端损失大小的水平不同,其风险因素的来源及其大小和方向也是随之变化的。     

Adjexpectile风险测度的性质、优化及资产配置的应用

金融风险的度量和识别是风险管理的重要内容,常用的风险度量工具是标准差、VaR、ES,但存在很多缺陷,expectile的提出弥补了这些不足,在理论界得到广泛的讨论和应用。本文扩展了expectile进行资产配置,提出Adjexpectile的概念,并讨论和分析了Adjexpectile的一致性风险度量、随机占优性、凸性,与标准差、VaR、shortfall的关系,风险贡献及风险分解的性质。通过对六个资产指数：上证国债指数、上证企业债指数、上证180指数、深圳100指数、深成长40p指数和黄金现货指数的复合周收益率数据进行组合优化配置,发现Adjexpectile在非对称性收益数据、组合前沿、风险分散方面具有一定的优越性。     

Statistical Inference for Expectile‐based Risk Measures

Expectiles were introduced by Newey and Powell in 1987 in the context of linear regression models. Recently, Bellini et al. revealed that expectiles can also be seen as reasonable law‐invariant risk measures. In this article, we show that the corresponding statistical functionals are continuous w.r.t. the 1‐weak topology and suitably functionally differentiable. By means of these regularity results, we can derive several properties such as consistency, asymptotic normality, bootstrap consistency and qualitative robustness of the corresponding estimators in nonparametric and parametric statistical models.     

Improved kth power expectile regression with nonignorable dropouts

The kth ( 1<k≤2) power expectile regression (ER) can balance robustness and effectiveness between the ordinary quantile regression and ER simultaneously. Motivated by a longitudinal ACTG 193A data with nonignorable dropouts, we propose a two-stage estimation procedure and statistical inference methods based on the kth power ER and empirical likelihood to accommodate both the within-subject correlations and nonignorable dropouts. Firstly, we construct the bias-corrected generalized estimating equations by combining the kth power ER and inverse probability weighting approaches. Subsequently, the generalized method of moments is utilized to estimate the parameters in the nonignorable dropout propensity based on sufficient instrumental estimating equations. Secondly, in order to incorporate the within-subject correlations under an informative working correlation structure, we borrow the idea of quadratic inference function to obtain the improved empirical likelihood procedures. The asymptotic properties of the corresponding estimators and their confidence regions are derived. The finite-sample performance of the proposed estimators is studied through simulation and an application to the ACTG 193A data is also presented.     

A new estimation method for continuous threshold expectile model

The continuous threshold expectile regression model could capture the effect of a covariate on the response variable with two different straight lines, while intersecting an unknown threshold needed be estimated. This article proposes a new estimation method via a linearization technique to estimate the regression coefficients and the threshold simultaneously. Statistical inferences of the proposed estimators are easily derived from the existing theory. Moreover, the estimation procedure is readily implemented by the current software. Simulation studies and an application on GDP per capita and quality of electricity supply data illustrate the proposed method.     

基于Expectile回归的均值-ES组合投资决策

为解决均值-ES（Expected Shortfall）组合投资决策中的计算困难，通过理论证明将其转化为一个Expectile回归问题，进而给出其Expectile回归求解新方法。该方法具有两个方面的优势：第一，Expectile回归的目标函数为二次损失函数，具有连续、光滑等特性，其优化与计算过程简单易行，且具有很好的可扩展性；第二，优化Expectile回归目标函数得到Expectile，利用Expectile与ES之间对应关系，能够准确地得到最优组合投资的ES风险值。选取沪深300指数中具有行业代表性的5支股票进行实证研究，将基于Expectile回归的均值-ES模型与均值-VaR模型、均值-方差模型进行对比，发现前者能够很好地分散组合投资尾部风险大小，显著提高组合投资绩效。     

条件自回归expectile模型及其在基金业绩评价中的应用

本文研究了日收益率之下开放式基金的业绩评价和检验问题,提出了改进的条件自回归expectile（CARE）模型并应用到基金业绩评价的问题研究中。首先运用非对称最小二乘法（ALS）对动态的CARE模型进行半参数估计,得到样本基金收益率序列的VaR值和ES值。其次,使用计算结果对样本基金的日收益率进行风险调整,得到基于VaR和ES修正的Sharpe比率。最后,在实证研究中,本文使用传统的Sharpe比率、基于VaR和ES的Sharpe比率对我国56只开放式基金在2005-2011年间的业绩进行了实证分析,结论显著证明了CARE模型在极端风险度量上更精确,在基金评价和检验中的应用中是可行的。     

基于非参多元Expectile模型的原油价格风险测度研究—宏观不确定性视角

当下政治经济环境存在诸多不确定性，原油价格随着不确定性的增加而大幅波动，因此在当前不确定性环境中建立一个有效的风险预测模型具有重要的实际意义。本文基于非参多元Expectile模型，选取2010年1月5日至2020年1月6日的美国西德克萨斯原油价格的日度数据，构建同时包含地缘政治风险、经济政策不确定性等六个宏观不确定性变量的原油价格风险预测模型。此外，引入APARCH模型和基于蒙特卡罗方法的GARCH模型，比较以上三个模型预测能力。最后，基于预测的VaR值计算调整的Sharpe比率。结论表明，整体上，非参多元Expectile模型能较好处理多个宏观变量包含的信息，具有更高的预测能力。在不确定性事件叠加发生的时期预测表现依然优于其他模型，减少了不确定性增加导致原油市场波动幅度增加带来的风险，具有更强的稳定性。因此，在经济转型的关键时期，本研究可为政策制定者和监管当局面临不确定性上升环境下建立有效的原油价格风险预测模型提供参考，制定应对政策防范化解风险，同时也为投资者在当前复杂的国际形势下提供预测参考，尽量规避损失同时获取收益。