A Markov Chain Copula Model for Credit Default Swaps with Bilateral Counterparty Risk

Reduced-form credit risk models are widely used in pricing and hedging credit derivatives. Generating default dependency is the key element in any such model. In this article, we use Markov copulae approach to model the dependence structure of defaults between the three obligors, one is the reference entity, another is the protection seller, the other is the protection buyer(the investor), so we can consider the bilateral counterparty risk of credit default swaps(CDS). In this Markov chain copula model, we obtain the explicit formulas of the CDS premium rates C ₁(T) (with unilateral counterparty risk) and C ₂(T) (with bilateral counterparty risk). And then we perform some numerical experiments to analyze the difference of the fair spreads between the unilateral case and the bilateral case.     

Archimedean copulae for risk measurement

In this paper some Archimedean copula functions for bivariate financial returns are studied. The choice of this family is due to their ability to capture the tail dependence, which is an association measure we can detect in many bivariate financial time-series. A time-varying version of these copulae is also investigated. Finally, the Value-at-Risk is computed and its performance is compared across different copula specifications.     

Non-parametric Estimation of Tail Dependence

Abstract.  Dependencies between extreme events (extremal dependencies) are attracting an increasing attention in modern risk management. In practice, the concept of tail dependence represents the current standard to describe the amount of extremal dependence. In theory, multi-variate extreme-value theory turns out to be the natural choice to model the latter dependencies. The present paper embeds tail dependence into the concept of tail copulae which describes the dependence structure in the tail of multivariate distributions but works more generally. Various non-parametric estimators for tail copulae and tail dependence are discussed, and weak convergence, asymptotic normality, and strong consistency of these estimators are shown by means of a functional delta method. Further, weak convergence of a general upper-order rank-statistics for extreme events is investigated and the relationship to tail dependence is provided. A simulation study compares the introduced estimators and two financial data sets were analysed by our methods.     

Inhomogeneous Dependence Modeling with Time-Varying Copulae

Measuring dependence in multivariate time series is tantamount to modeling its dynamic structure in space and time. In risk management, the nonnormal behavior of most financial time series calls for non-Gaussian dependences. The correct modeling of non-Gaussian dependences is, therefore, a key issue in the analysis of multivariate time series. In this article we use copula functions with adaptively estimated time-varying parameters for modeling the distribution of returns. Furthermore, we apply copulae to the estimation of Value-at-Risk of portfolios and show their better performance over the RiskMetrics approach.     

Accuracy of variational estimates for random graph mixture models

Variational and variational Bayes techniques are popular approaches for statistical inference of complex models but their theoretical properties are still not well known. Because of both unobserved variables and intricate dependency structures, mixture models for random graphs constitute a good case study. We first present four different variational estimates for the parameters of these models. We then compare their accuracy through simulation studies and show that the variational Bayes estimates seem the most accurate for moderate graph size. We finally re-analyse the regulatory network of Escherichia coli with this approach.     

Estimators based on trimmed Kendall’s tau in multivariate copula models

A common method of estimating the parameters of dependency in multivariate copula models is by maximum likelihood principle, termed as Inference From Marginals (IFM); see Joe (1997)  [13]. To avoid possible misspecification of the marginal distributions, some authors suggest rank-based procedures for estimating the parameters of dependency in a multivariate copula model. A standard approach for this problem is through maximization of the pseudolikelihood, as discussed in Genest et al. (1995)  [9] and Shih and Louis (1995)  [23]. Alternative estimators based on the inversion of two multivariate extensions of Kendall’s tau, due to Kendall and Babington Smith (1940)  [14] and Joe (1990)  [12], were used in Genest et al. (2011)  [10]. In the literature, dependency of data was considered in the whole data space. However, it may be better to divide the data set into two distinct sets, lower and higher than a threshold, and then evaluate the dependency parameters in these sets. In this way, we may have different dependency parameters in these sets which may shed additional light. For example, in drought analysis, precipitation and minimum temperature may be modeled using copulas in which case we can infer that dependency between precipitation and minimum temperature are severe when they are less than a certain threshold. In this paper, after introducing trimmed Kendall’s tau when such a threshold is imposed, we consider modeling dependency using it as a measure. Asymptotic distribution of trimmed Kendall’s tau is also investigated, and a test for the null hypothesis of equality between Kendall’s tau and trimmed Kendall’s tau is constructed. We can use this hypothesis testing procedure for testing the hypothesis that data are dependent before a threshold value and are independent after the threshold. An explicit form of the asymptotic distribution of trimmed Kendall’s tau and of the mentioned test statistic are also derived for some special families of copulas. Finally, the results of a simulation study and an illustrative example are provided.     

基于深度学习LSTM神经网络的全球股票指数预测研究

作为深度学习技术的经典模型之一，长短期记忆（LSTM）神经网络在挖掘序列数据长期依赖关系中极具优势。基于深度神经网络优化技术，本文构造了一个深层LSTM神经网络并将其应用于全球30个股票指数三种不同期限的预测研究，结果发现：①LSTM神经网络具有很强的泛化能力，对全部指数不同期限的预测效果均很稳定；②LSTM神经网络具有优秀的预测精度，相比三种对照模型（SVR，MLP和ARIMA），其对全部指数的平均预测精度在不同期限上均有提升；③LSTM神经网络能够有效控制误差波动，其对全部指数的平均预测稳定度相比三种对照模型在不同期限上亦均有提高。鉴于LSTM神经网络在预测精度和稳定度两方面的优势，其未来在金融预测中将有广阔的应用前景。     

The alert-delay competing risks model for maintenance analysis

The paper considers the modelling of the dependency between corrective maintenance and condition-based preventive maintenance of complex repairable systems. A new model of dependent competing risks is proposed, called the alert-delay (AD) model. This model has different properties from that of the delay-time, repair-alert and proportional warning constant inspection models and happens to fit some data sets which could not be fitted by the previous models. The main features of the AD model are derived: probabilistic properties and statistical analysis. Simulation results and an application to real data are presented.     

Logistic与分类树模型变量筛选的比较——基于信用卡邮寄业务响应率分析

基于信用卡邮寄业务响应率分析来讨论Logistic模型和分类树模型在变量选取上的区别,并尝试从几个不同角度去解释两类模型变量筛选差异的原因。笔者认为没有绝对占优势的方法,需要结合具体场景和模型的特点来选择合适的模型。分类树模型在训练集上容易过度拟合,对单个变量的影响很敏感,在进行危险因素分析时结果更能强调危险因素,对孤立点的识别率很高。Logistic模型容易受到解释变量依存关系的影响,加上分类变量的影响容易过多地选入变量或者因子,对孤立点敏感,对噪点不敏感。判别函数的差异是变量筛选差异的关键因素。     

Estimation of the Force of Infection from Current Status Data Using Generalized Linear Mixed Models

Based on sero-prevalence data of rubella, mumps in the UK and varicella in Belgium, we show how the force of infection, the age-specific rate at which susceptible individuals contract infection, can be estimated using generalized linear mixed models (McCulloch & Searle, 2001). Modelling the dependency of the force of infection on age by penalized splines, which involve fixed and random effects, allows us to use generalized linear mixed models techniques to estimate both the cumulative probability of being infected before a given age and the force of infection. Moreover, these models permit an automatic selection of the smoothing parameter. The smoothness of the estimated force of infection can be influenced by the number of knots and the degree of the penalized spline used. To determine these, a different number of knots and different degrees are used and the results are compared to establish this sensitivity. Simulations with a different number of knots and polynomial spline bases of different degrees suggest - for estimating the force of infection from serological data - the use of a quadratic penalized spline based on about 10 knots.     

Joint coverage probability in a simulation study on continuous-time Markov chain parameter estimation

Parameter dependency within data sets in simulation studies is common, especially in models such as continuous-time Markov chains (CTMCs). Additionally, the literature lacks a comprehensive examination of estimation performance for the likelihood-based general multi-state CTMC. Among studies attempting to assess the estimation, none have accounted for dependency among parameter estimates. The purpose of this research is twofold: (1) to develop a multivariate approach for assessing accuracy and precision for simulation studies (2) to add to the literature a comprehensive examination of the estimation of a general 3-state CTMC model. Simulation studies are conducted to analyze longitudinal data with a trinomial outcome using a CTMC with and without covariates. Measures of performance including bias, component-wise coverage probabilities, and joint coverage probabilities are calculated. An application is presented using Alzheimer's disease caregiver stress levels. Comparisons of joint and component-wise parameter estimates yield conflicting inferential results in simulations from models with and without covariates. In conclusion, caution should be taken when conducting simulation studies aiming to assess performance and choice of inference should properly reflect the purpose of the simulation.     

Overview of current population trends in the Federal Republic of Germany and prospects for the future

Current population trends in the Federal Republic of Germany are first reviewed. Topics discussed include demographic aging, ethnic composition, declining fertility, and households. Population projections to the year 2030 are then presented by age group using several different models. Finally, the economic and social policy consequences of long-term population trends are examined, with particular reference to the economically active population and the dependency burden.     

Stochastic population forecasts for the United States: beyond high,medium, and low

"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."     

Moving average models—time series in m-dimensions

Stochastic models for discrete time series in the time domain are well known but such models lack consideration of spatial dependency I We expand on their work by constructing spatially dependent moving average models. Definitions of order, stationarity, invertibility, autocorrelation function, and spectrum are made as natural extensions of those in zero dimensions and are implemented in the one and two-space dimensional models.     

Score tests for independence in semiparametric competing risks models

A popular model for competing risks postulates the existence of a latent unobserved failure time for each risk. Assuming that these underlying failure times are independent is attractive since it allows standard statistical tools for right-censored lifetime data to be used in the analysis. This paper proposes simple independence score tests for the validity of this assumption when the individual risks are modeled using semiparametric proportional hazards regressions. It assumes that covariates are available, making the model identifiable. The score tests are derived for alternatives that specify that copulas are responsible for a possible dependency between the competing risks. The test statistics are constructed by adding to the partial likelihoods for the individual risks an explanatory variable for the dependency between the risks. A variance estimator is derived by writing the score function and the Fisher information matrix for the marginal models as stochastic integrals. Pitman efficiencies are used to compare test statistics. A simulation study and a numerical example illustrate the methodology proposed in this paper.     

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson (GP) distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding correlated spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. in Stat Comput 12(4):353–367, (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. in J R Stat Soc B64(4):583–640, (2002) and using proper scoring rules, see for example Gneiting and Raftery in Technical Report no. 463, University of Washington, (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, spatial Poisson models with spatially correlated or uncorrelated random effects are to be preferred over all other models according to the considered criteria.     

Survival estimation and the effects of dependency among animals

Survival models assume that fates of individuals are independent, yet the robustness of this assumption has been poorly quantified. We examine how empirically derived estimates of the variance of survival rates are affected by dependency in survival probability among individuals. We used Monte Carlo simulations to generate known amounts of dependency among pairs of individuals and analyzed these data with Kaplan-Meier and Cormack-Jolly-Seber models. Dependency significantly increased these empirical variances as compared to theoretically derived estimates of variance from the same populations. Using resighting data from 168 pairs of black brant ( Branta bernicla nigricans ), we used a resampling procedure and program RELEASE to estimate empirical and mean theoretical variances. We estimated that the relationship between paired individuals caused the empirical variance of the survival rate to be 155% larger than the empirical variance for unpaired individuals. Monte Carlo simulations and use of this resampling strategy can provide investigators with information on how robust their data are to this common assumption of independent survival probabilities.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

金融发展与贸易依存度的实证研究——基于东部十省市面板数据的协整分析

本文利用东部十省市的面板数据,研究金融发展与贸易依存度的关系。运用面板单位根检验和面板协整检验,建立面板协整模型进行分析。结果显示：金融发展仅与进口依存度存在长期均衡关系,且不同省市金融发展对进口依存度的推动作用不同。最后分析其作用差异的潜在原因,并根据实证结果提出了政策建议。     

Survival estimation and the effects of dependency among animals

Survival models assume that fates of individuals are independent, yet the robustness of this assumption has been poorly quantified. We examine how empirically derived estimates of the variance of survival rates are affected by dependency in survival probability among individuals. We used Monte Carlo simulations to generate known amounts of dependency among pairs of individuals and analyzed these data with Kaplan-Meier and Cormack-Jolly-Seber models. Dependency significantly increased these empirical variances as compared to theoretically derived estimates of variance from the same populations. Using resighting data from 168 pairs of black brant ( Branta bernicla nigricans ), we used a resampling procedure and program RELEASE to estimate empirical and mean theoretical variances. We estimated that the relationship between paired individuals caused the empirical variance of the survival rate to be 155% larger than the empirical variance for unpaired individuals. Monte Carlo simulations and use of this resampling strategy can provide investigators with information on how robust their data are to this common assumption of independent survival probabilities.