Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case

Abstract. In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high‐dimensional vector after standardizing marginals. Hence, it is of importance to model and estimate tail dependence functions. Even for moderate dimension, non‐parametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper, we propose a semi‐parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Under this model assumption, we propose a novel estimator for the tail dependence function, which proves favourable compared to the empirical tail dependence function estimator, both theoretically and empirically.     

Copula Density Estimation by Total Variation Penalized Likelihood

Copulas are full measures of dependence among random variables. They are increasingly popular among academics and practitioners in financial econometrics for modeling comovements between markets, risk factors, and other relevant variables. A copula's hidden dependence structure that couples a joint distribution with its marginals makes a parametric copula non-trivial. An approach to bivariate copula density estimation is introduced that is based on a penalized likelihood with a total variation penalty term. Adaptive choice of the amount of regularization is based on approximate Bayesian Information Criterion (BIC) type scores. Performance are evaluated through the Monte Carlo simulation.     

Risk analysis in the brazilian stock market: copula-APARCH modeling for value-at-risk

Risk management of stock portfolios is a fundamental problem for the financial analysis since it indicates the potential losses of an investment at any given time. The objective of this study is to use bivariate static conditional copulas to quantify the dependence structure and to estimate the risk measure Value-at-Risk (VaR). There were selected stocks that have been performing outstandingly on the Brazilian Stock Exchange to compose pairs trading portfolios (B3, Gerdau, Magazine Luiza, and Petrobras). Due to the flexibility that this methodology offers in the construction of multivariate distributions and risk aggregation in finance, we used the copula-APARCH approach with the Normal, T-student, and Joe-Clayton copula functions. In most scenarios, the results showed a pattern of dependence at the extremes. Moreover, the copula form seems not to be relevant for VaR estimation, since in most portfolios the appropriate copulas lead to significant VaR estimates. It has found that the best models fitted provided conservative risk measures, estimates at 5% and 1%, in a scenario more aggressive.     

Statistical properties of parametric estimators for Markov chain vectors based on copula models

To estimate and measure risks, two key classes of dependence relationship must be identified: temporal dependence and contemporaneous dependence. In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE), and we investigate the consistency and asymptotic normality of parametric estimators. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio under the 3SPMLE for stationary Markov vector-type models.     

A method for constructing asymmetric pair-copula and its application

Pair-copula has become a hot spot in the research of both theory and application of statistics. Therefore, proper construction of pair-copula is an area worthy of study. Considering the asymmetry of variate dependence in practical applications and the disadvantages of the widely used asymmetric copulas, we propose a method to construct asymmetric pair-copula. In our method, we treat the asymmetric bivariate copula constructed by Liebscher's method as a generator, using this generator to construct asymmetric pair-copula. Also, on the basis of our method, we propose and prove a reference for selecting copula family in the construction. To verify the method, we construct asymmetric copulas and asymmetric pair-copulas using the daily runoff data collected at Yichang hydrological station to describe the extreme drought events of Yangtze River. After comparing the models in some aspects, we accept a model we construct, and the result displays the feasibility and practicality of the method we propose.     

相依结构、动态系统性风险测度与后验分析

本文利用Copula相依结构理论扩展和求解了现有的系统性风险测度CoVaR,以得到适用于不同类型常参数和时变参数Copula函数及不同分布假设的动态系统性风险测度。为了验证和评估模型设定的准确性与应用价值,我们构建了适用于该动态系统性风险测度CoVaR的严谨后验分析工具。除“无条件覆盖性”、“独立性”和“条件覆盖性”外,我们首次提出了“混合独立性”检验。基于中国14家上市商业银行的实证分析表明：中国上市商业银行与中国银行业之间的相依结构呈现多样化特征;无论是样本内还是样本外预测区间,我们的动态Copula-CoVaR模型能够有效地捕捉典型系统性风险事件;严谨的后验分析不仅需要检验系统性风险测度CoVaR,也需要检验条件事件的临界值VaR。     

Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution

Value at Risk (VaR) forecasts can be produced from conditional autoregressive VaR models, estimated using quantile regression. Quantile modeling avoids a distributional assumption, and allows the dynamics of the quantiles to differ for each probability level. However, by focusing on a quantile, these models provide no information regarding expected shortfall (ES), which is the expectation of the exceedances beyond the quantile. We introduce a method for predicting ES corresponding to VaR forecasts produced by quantile regression models. It is well known that quantile regression is equivalent to maximum likelihood based on an asymmetric Laplace (AL) density. We allow the density's scale to be time-varying, and show that it can be used to estimate conditional ES. This enables a joint model of conditional VaR and ES to be estimated by maximizing an AL log-likelihood. Although this estimation framework uses an AL density, it does not rely on an assumption for the returns distribution. We also use the AL log-likelihood for forecast evaluation, and show that it is strictly consistent for the joint evaluation of VaR and ES. Empirical illustration is provided using stock index data. Supplementary materials for this article are available online.     

The copula directional dependence by stochastic volatility models

This paper proposes a copula directional dependence by using a bivariate Gaussian copula beta regression with Stochastic Volatility (SV) models for marginal distributions. With the asymmetric copula generated by the composition of two Plackett copulas, we show that our SV copula directional dependence by the Gaussian copula beta regression model is superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of the percent relative efficiency of bias and mean squared error. To validate our proposed method with the real data, we use Brent Crude Daily Price (BRENT), West Texas Intermediate Daily Price (WTI), the Standard & Poor’s 500 (SP) and US 10-Year Treasury Constant Maturity Rate (TCM) so that our copula SV directional dependence is overall superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of precision by the percent relative efficiency of mean squared error. In terms of forecasting using the real financial data, we also show that the Bayesian SV model of the uniform transformed data by a copula conditional distribution yields an improvement on the volatility models such as GARCH and SV.     

Tail-weighted measures of dependence

Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall's tau and Spearman's rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures – the portfolio value-at-risk and conditional tail expectation.     

A quasi-Bayesian model averaging approach for conditional quantile models

The value at risk (VaR) is a risk measure that is widely used by financial institutions to allocate risk. VaR forecast estimation involves the evaluation of conditional quantiles based on the currently available information. Recent advances in VaR evaluation incorporate conditional variance into the quantile estimation, which yields the conditional autoregressive VaR (CAViaR) models. However, uncertainty with regard to model selection in CAViaR model estimators raises the issue of identifying the better quantile predictor via averaging. In this study, we propose a quasi-Bayesian model averaging method that generates combinations of conditional VaR estimators based on single CAViaR models. This approach provides us a basis for comparing single CAViaR models against averaged ones for their ability to forecast VaR. We illustrate this method using simulated and financial daily return data series. The results demonstrate significant findings with regard to the use of averaged conditional VaR estimates when forecasting quantile risk.     

系统性风险度量CoES的建模和检验

本文结合极值理论(Extreme Value Theory,EVT)和新的动态混合Copula(Dynamic Mixture Copula,DM-Copula)函数,提出了一种新的CoES估计方法DM-Copula-EVT。在EVT建模中,本文改进了阈值的选取方法以避免选择的主观性,并提出了一系列新的动态混合Copula以更好地刻画金融市场日益复杂的尾部关联性。此外,本文首次提出了检验CoES模型设定正确性的后验分析方法,包括无条件覆盖性检验和条件覆盖性检验。将本文建模和检验方法应用于我国金融市场,研究发现:相对于传统使用的t分布,EVT能更好地拟合指数的尾部分布;新的动态混合Copula函数能更好地刻画金融部门与系统之间的复杂关联性。     

Conditional Euro Area Sovereign Default Risk

We propose an empirical framework to assess the likelihood of joint and conditional sovereign default from observed CDS prices. Our model is based on a dynamic skewed-t distribution that captures all salient features of the data, including skewed and heavy-tailed changes in the price of CDS protection against sovereign default, as well as dynamic volatilities and correlations that ensure that uncertainty and risk dependence can increase in times of stress. We apply the framework to euro area sovereign CDS spreads during the euro area debt crisis. Our results reveal significant time-variation in distress dependence and spill-over effects for sovereign default risk. We investigate market perceptions of joint and conditional sovereign risk around announcements of Eurosystem asset purchases programs, and document a strong impact on joint risk.     

基于Copula方法的国债市场相依风险度量

本文讨论了如何利用Copula连接函数对多元金融数据的相依结构进行统计建模,首先对几种常用的Copula连接函数进行了介绍,分析了不同边际分布和不同Copula函数的选取对联合分布产生的影响,然后讨论了Copula函数的选取和其参数的估计问题,最后利用我国国债数据进行实证分析,得到了不同组合的风险值。     

Production Frontiers and Panel Data

Value at risk (VaR) is the standard measure of market risk used by financial institutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, the conditional autoregressive value at risk (CAViaR) model specifies the evolution of the quantile over time using an autoregressive process and estimates the parameters with regression quantiles. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the dynamic quantile test. Applications to real data provide empirical support to this methodology.     

Estimating non-simplified vine copulas using penalized splines

Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so-called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first nonparametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback–Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.     

Statistical Modeling of Temporal Dependence in Financial Data via a Copula Function

In financial analysis it is useful to study the dependence between two or more time series as well as the temporal dependence in a univariate time series. This article is concerned with the statistical modeling of the dependence structure in a univariate financial time series using the concept of copula. We treat the series of financial returns as a first order Markov process. The Archimedean two-parameter BB7 copula is adopted to describe the underlying dependence structure between two consecutive returns, while the log-Dagum distribution is employed to model the margins marked by skewness and kurtosis. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. Furthermore, we apply the model to the daily returns of four stocks and, finally, we illustrate how its fitting to data can be improved when the dependence between consecutive returns is described through a copula function.     

CVaR-EVT和BMM在极端金融风险管理中的应用研究

随着风险度量一致性原则的提出,研究发现金融机构广泛采用的VaR模型存在严重不足,尤其针对分布具有厚尾特征的极端金融风险无法有效度量。本文采用极值理论（EVT）解决VaR方法的尾部度量不足问题,利用CVaR-EVT和BMM模型分析美国、香港股票市场和我国沪深两市指数18年的日收益数据,研究发现：（1）在95％置信区间及点估计中,分位数为99%的CVaR-EVT所揭示的极端风险优于VaR的估计值;且BMM方法为实施长期极端风险管理提供了有力决策依据,其回报率受分段时区的影响,期间越长,风险估计值越高;（2）模型采用ML和BS方法统计估值显示,我国股票市场极端风险尾部估计值高于香港和美国市场;但是,国内市场逐步稳定,并呈现出跟进国际市场且差距缩小的发展趋势。     

基于Pair-Copula贝叶斯网络模型的金融危机传播效应研究

为克服传统Copula函数建模的复杂性,采用Pair-Copula贝叶斯网络方法,将Copula函数与贝叶斯网络的优势相结合,有效降低了高维Copula函数参数估计的复杂程度,解决了连续型贝叶斯网络非正态下边际分布难以控制的问题,有利于捕捉网络结点间的非对称相依结构,从而构建了国际8个代表性股票市场在次贷危机爆发前、中、后各个时期的概率网络模型,对危机在主要国际市场上的传播效应问题及传播路径识别问题进行实证研究。研究结果表明:金融危机在国际金融市场上具有明显的区域性传播特征;香港市场在金融危机发生前后一直都是亚洲市场的连接枢纽,是风险防范的关键节点;危机发生后欧洲市场与亚洲市场间的紧密程度加强;金融危机减缓了全球一体化的进程,但经济全球化的趋势势不可挡。     

投资组合动态VaR风险度量

投资组合的VaR风险度量依赖于投资组合中金融资产间联合分布函数的确定，随着投资组合规模的扩大，其VaR的计算难度也不断加大。利用ICA可以将多元联合概率分布函数转化为一元概率分布函数乘积实现简化计算的特点，基于ICA的投资组合动态VaR风险度量方法和计算步骤，克服了多元非正态条件下VaR测算上的困难。实证研究表明，与EWMA模型法、MGARCH模型法相比，ICA法能够准确地度量投资组合动态VaR。