Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence

We consider nonparametric estimation of multivariate versions of Blomqvist’s beta, also known as the medial correlation coefficient. For a two-dimensional population, the sample version of Blomqvist’s beta describes the proportion of data which fall into the first or third quadrant of a two-way contingency table with cutting points being the sample medians. Asymptotic normality and strong consistency of the estimators are established by means of the empirical copula process, imposing weak conditions on the copula. Though the asymptotic variance takes a complicated form, we are able to derive explicit formulas for large families of copulas. For the copulas of elliptically contoured distributions we obtain a variance stabilizing transformation which is similar to Fisher’s z-transformation. This allows for an explicit construction of asymptotic confidence bands used for hypothesis testing and eases the analysis of asymptotic efficiency. The computational complexity of estimating Blomqvist’s beta corresponds to the sample size n, which is lower than the complexity of most competing dependence measures.      

On the structure and estimation of hierarchical Archimedean copulas

In this paper we provide a method for estimating multivariate distributions defined through hierarchical Archimedean copulas. In general, the true structure of the hierarchy is unknown, but we develop a computationally efficient technique to determine it from the data. For this purpose we introduce a hierarchical estimation procedure for the parameters and provide an asymptotic analysis. We consider both parametric and nonparametric estimation of the marginal distributions. A simulation study and an empirical application show the effectiveness of the grouping procedure in the sense of structure selection.     

中国大陆与其主要贸易伙伴股市间的极值关联性

本文运用尾部相依系数作为度量极值关联性的工具,首先研究中国加入世界贸易组织后大陆股市与七个主要贸易伙伴股市的极值关联性强弱,即股市同时极端下跌的风险性大小;其次研究中国加入世界贸易组织前后大陆股市与世界股市整体的极值关联性变化情况。结果发现：相比欧美股市,中国大陆股市与亚洲股市联系更紧密,同时发生极端下跌的可能性更大。整体而言,中国大陆股市与世界其他股市配置资产已无法完全分散极值风险。因此,若投资者在中国大陆股市进行投资的同时也寻求其他国家或地区股市,为了降低极值风险发生的概率,应选择欧美国家的股市。     

Percentage Points of the Multivariate t Distribution

The known methods for computing percentage points of multivariate t distributions are reviewed. We believe that this review will serve as an important reference and encourage further research activities in the area.     

Multivariate survival functions characterized by constant product of mean remaining lives and hazard rates

Under the assumption that the products of multivariate mean remaining lives and hazard rates are the same constant, it is shown that the corresponding multivariate survival function belongs to one of three families: (1) multivariate Gumbel exponential distribution; (2) multivariate Lomax (Pareto type II) distribution; (3) multivariate rescaled Dirichlet distribution. This result is then used to derive another characterization of the latter two families based on the residual life distribution.     

Characteristic properties of multivariate survival functions in terms of residual life distributions

This paper discusses the relationships among some characteristic properties of the multivariate survival function based on the residual life distribution, and provides the conditions for their equivalence. In the meanwhile, the corrected version of Ma (1996, Theorem 1 (ii) and (iii)) is given.     

利用GARCH-M类模型和极值理论对上证综指的研究

金融时间序列具有分布的厚尾性、波动的集聚性等特征,传统的方法难以准确的度量其风险。文中运用一种新的估计VaR和ES的方法,即采取两阶段法。首先用GARCH-M类模型(GARCH-M、EGARCH-M和TGARCH-M)拟和原始收益率数据,得到残差序列;第二步用极值分析的方法分析的尾部,最后得到收益率序列的动态VaR和ES。最后对三个模型的计算结果进行比较。     

Portfolio optimization based on GARCH-EVT-Copula forecasting models

This study uses GARCH-EVT-copula and ARMA-GARCH-EVT-copula models to perform out-of-sample forecasts and simulate one-day-ahead returns for ten stock indexes. We construct optimal portfolios based on the global minimum variance (GMV), minimum conditional value-at-risk (Min-CVaR) and certainty equivalence tangency (CET) criteria, and model the dependence structure between stock market returns by employing elliptical (Student-$t$ and Gaussian) and Archimedean (Clayton, Frank and Gumbel) copulas. We analyze the performances of 288 risk modeling portfolio strategies using out-of-sample back-testing. Our main finding is that the CET portfolio, based on ARMA-GARCH-EVT-copula forecasts, outperforms the benchmark portfolio based on historical returns. The regression analyses show that GARCH-EVT forecasting models, which use Gaussian or Student-$t$ copulas, are best at reducing the portfolio risk.     

基于极值理论的资产组合风险研究

在金融系统中风险管理者十分关注投资风险的大小,尤其是在极端情况下的风险大小。市场风险值()是一种常用的度量风险的方法。本文将极值理论用于中国上证指数和深成指数市场风险值的度量,同时探讨了用极值理论评价资产组合风险的方法,并将其计算结果与基于正态分布同t分布的方法进行比较,发现采用极值理论度量市场风险值要优于经典的方法。     

An Extreme Value Approach to Test the Effect of Price Limits on Volatility

Many stock exchanges around the world enforcing daily price limits on the amount asset prices can change to prevent the market from overreacting and to reduce volatility. Price limits are artificial boundaries set by market regulators who restrict price changes of a stock to a pre-specified range during a trading day or a single trading session. The primary aim of price limit rules is to stabilize the markets during panic trading, to moderate vitality by repressing excessive speculation, and to allow stocks to be traded at prices close to their fair value. However, their impact on the market is a somewhat unresolved issue （Harris, 1998）. Using a methodology of comparing volatility based on the extreme value technique, the authors empirically investigate the impact of price limits on the volatility of the Stock Exchange of Thailand. The empirical results support price limits advocates, suggesting that price limits rules moderate stock price volatility.     

Abnormal returns, risk, and options in large data sets

Large data sets in finance with millions of observations have become widely available. Such data sets enable the construction of reliable semi-parametric estimates of the risk associated with extreme price movements. Our approach is based on semi-parametric statistical extreme value analysis, and compares favorably with the conventional finance normal distribution based approach. It is shown that the efficiency of the estimator of the extreme returns may benefit from high frequency data. Empirical tail shapes are calculated for the German Mark—US Dollar foreign exchange rate, and we use the semi-parametric tail estimates in combination with the empirical distribution function to evaluate the returns on exotic options.     

The emperor's new clothes: a critique of the multivariate t regression model

Zellner (1976) proposed a regression model in which the data vector (or the error vector) is represented as a realization from the multivariate Student t distribution. This model has attracted considerable attention because it seems to broaden the usual Gaussian assumption to allow for heavier-tailed error distributions. A number of results in the literature indicate that the standard inference procedures for the Gaussian model remain appropriate under the broader distributional assumption, leading to claims of robustness of the standard methods. We show that, although mathematically the two models are different, for purposes of statistical inference they are indistinguishable. The empirical implications of the multivariate t model are precisely the same as those of the Gaussian model. Hence the suggestion of a broader distributional representation of the data is spurious, and the claims of robustness are misleading. These conclusions are reached from both frequentist and Bayesian perspectives.     

极端波动情景中的压力测试和极值理论方法研究

VaR描述的是市场正常波动情况下的资产组合最大可能损失,指出了不利事件发生的概率,但没有说明不利事件发生时的实际损失到底有多大。为了测量在这些小概率极端情况下的风险,本文对当前测量极端波动情景中较为先进的情景分析、系统化压力测试和极值分析方法进行较为全面的研究。