Modeling spot price dependence in Australian electricity markets with applications to risk management

We examine the dependence structure of electricity spot prices across regional markets in Australia. One of the major objectives in establishing a national electricity market was to provide a nationally integrated and efficient electricity market, limiting market power of generators in the separate regional markets. Our analysis is based on a GARCH approach to model the marginal price series in the considered regions in combination with copulae to capture the dependence structure between the marginals. We apply different copula models including Archimedean, elliptical and copula mixture models. We find a positive dependence structure between the prices for all considered markets, while the strongest dependence is exhibited between markets that are connected via interconnector transmission lines. Regarding the nature of dependence, the Student-t copula provides a good fit to the data, while the overall best results are obtained using copula mixture models due to their ability to also capture asymmetric dependence in the tails of the distribution. Interestingly, our results also suggest that for the four major markets, NSW, QLD, SA and VIC, the degree of dependence has decreased starting from the year 2008 towards the end of the sample period in 2010. Examining the Value-at-Risk of stylized portfolios constructed from electricity spot contracts in different markets, we find that the Student-t and mixture copula models outperform the Gaussian copula in a backtesting study. Our results are important for risk management and hedging decisions of market participants, in particular for those operating in several regional markets simultaneously.     

The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study

The effect on the estimation of the Value at Risk when dealing with multivariate portfolios when there is a misspecification both in the marginals and in the copulas is investigated. It is first shown that, when there is skewness in the data and symmetric marginals are used, the estimated elliptical (normal or t) copula correlations are negatively biased, reaching values as high as 70% of the true values. Besides, the bias almost doubles if negative correlations are considered, compared to positive correlations. As for the t copula degrees of freedom parameter, the use of wrong marginals delivers large positive biases, instead. If the dependence structure is represented by a copula which is not elliptical, e.g. the Clayton copula, the effects of marginal misspecifications on the copula parameter estimation can be rather different, depending on the sign of marginal skewness. Extensive Monte Carlo studies then show that the misspecifications in the marginal volatility equation more than offset the biases in copula parameters when VaR forecasting is of concern, small samples are considered and the data are leptokurtic. The biases in the volatility parameters are much smaller, whereas those ones in the copula parameters remain almost unchanged or even increase when the sample dimension increases. In this case, copula misspecifications do play a role for VaR estimation. However, these effects depend heavily on the sign of the dependence.     

Vine copulas for mixed data : multi-view clustering for mixed data beyond meta-Gaussian dependencies

Copulas enable flexible parameterization of multivariate distributions in terms of constituent marginals and dependence families. Vine copulas, hierarchical collections of bivariate copulas, can model a wide variety of dependencies in multivariate data including asymmetric and tail dependencies which the more widely used Gaussian copulas, used in Meta-Gaussian distributions, cannot. However, current inference algorithms for vines cannot fit data with mixed—a combination of continuous, binary and ordinal—features that are common in many domains. We design a new inference algorithm to fit vines on mixed data thereby extending their use to several applications. We illustrate our algorithm by developing a dependency-seeking multi-view clustering model based on Dirichlet Process mixture of vines that generalizes previous models to arbitrary dependencies as well as to mixed marginals. Empirical results on synthetic and real datasets demonstrate the performance on clustering single-view and multi-view data with asymmetric and tail dependencies and with mixed marginals.     

A new algorithm based on copulas for VaR valuation with empirical calculations

This paper concerns the application of copula functions in VaR valuation. The copula function is used to model the dependence structure of multivariate assets. After the introduction of the traditional Monte Carlo simulation method and the pure copula method we present a new algorithm based on mixture copula functions and the dependence measure, Spearman’s rho. This new method is used to simulate daily returns of two stock market indices in China, Shanghai Stock Composite Index and Shenzhen Stock Composite Index, and then empirically calculate six risk measures including VaR and conditional VaR. The results are compared with those derived from the traditional Monte Carlo method and the pure copula method. From the comparison we show that the dependence structure between asset returns plays a more important role in valuating risk measures comparing with the form of marginal distributions.     

Extremal financial risk models and portfolio evaluation

It is difficult to find an existing single model which is able to simultaneously model exceedances over thresholds in multivariate financial time series. A new modeling approach, which is a combination of max-stable processes, GARCH processes, and Markov processes, is proposed. Combining Markov processes and max-stable processes defines a new statistical model which has the flexibility of modeling cross-sectional tail dependencies between risk factors and tail dependencies across time. The new model also models asymmetric behaviors of negative and positive returns on financial assets. An important application of the proposed method is to calculate value at risk (VaR) and evaluate portfolio combinations under VaR constraints. Result comparisons between VaRs based on the new approach and VaRs based on some existing methods such as variance-covariance approach and historical simulation approach suggest that some existing methods substantially underestimate the risks during recession and expansion time.     

Quantile curves and dependence structure for bivariate distributions

Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points—level curve—which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.     

利用Copula函数估计概率模型并采样的分布估计算法

将Copula理论引入分布估计算法的研究中,并在估计概率模型时分两个步骤进行：1）估计各变量的边缘分布函数;2）构造经验copula函数或正态Copula函数．根据Copula函数和各边缘分布进行采样,在简化估计模型运算复杂度的同时,充分反映了变量之间的关系．仿真实验验证了该算法的可行性和有效性．     

A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

A Bayesian semi-parametric bivariate failure time model

In this paper we introduce a Bayesian semiparametric model for bivariate and multivariate survival data. The marginal densities are well-known nonparametric survival models and the joint density is constructed via a mixture. Our construction also defines a copula and the properties of this new copula are studied. We also consider the model in the presence of covariates and, in particular, we find a simple generalisation of the widely used frailty model, which is based on a new bivariate gamma distribution.     

Testing the bivariate distribution of daily equity returns using copulas. An application to the Spanish stock market

The problem of the identification of dependencies between time series of equity returns is analyzed. Marginal distribution functions are assumed to be known, and a bivariate chi-square test of fit is applied in a fully parametric copula approach. Several marginal models and families of copulas are fitted and compared with Spanish stock market data. The results show the difficulty in adjusting the bivariate distribution of raw returns, and highlight the effect of a GARCH filtering in the selection of the best fitting copula.     

Comparison of semiparametric maximum likelihood estimation and two-stage semiparametric estimation in copula models

We consider bivariate distributions that are specified in terms of a parametric copula function and nonparametric or semiparametric marginal distributions. The performance of two semiparametric estimation procedures based on censored data is discussed: maximum likelihood (ML) and two-stage pseudolikelihood (PML) estimation. The two-stage procedure involves less computation and it is of interest to see whether it is significantly less efficient than the full maximum likelihood approach. We also consider cases where the copula model is misspecified, in which case PML may be better. Extensive simulation studies demonstrate that in the absence of covariates, two-stage estimation is highly efficient and has significant robustness advantages for estimating marginal distributions. In some settings, involving covariates and a high degree of association between responses, ML is more efficient. For the estimation of association, PML does not offer an advantage.     

Semiparametric bivariate Archimedean copulas

While parametric copulas often lack expressive capacity to capture the complex dependencies that are usually found in empirical data, non-parametric copulas can have poor generalization performance because of overfitting. A semiparametric copula method based on the family of bivariate Archimedean copulas is introduced as an intermediate approach that aims to provide both accurate and robust fits. The Archimedean copula is expressed in terms of a latent function that can be readily represented using a basis of natural cubic splines. The model parameters are determined by maximizing the sum of the log-likelihood and a term that penalizes non-smooth solutions. The performance of the semiparametric estimator is analyzed in experiments with simulated and real-world data, and compared to other methods for copula estimation: three parametric copula models, two semiparametric estimators of Archimedean copulas previously introduced in the literature, two flexible copula methods based on Gaussian kernels and mixtures of Gaussians and finally, standard parametric Archimedean copulas. The good overall performance of the proposed semiparametric Archimedean approach confirms the capacity of this method to capture complex dependencies in the data while avoiding overfitting.     

Dependence tree structure estimation via copula

We propose an approach for dependence tree structure learning via copula. A nonparametric algorithm for copula estimation is presented. Then a Chow-Liu like method based on dependence measure via copula is proposed to estimate maximum spanning bivariate copula associated with bivariate dependence relations. The main advantage of the approach is that learning with empirical copula focuses on dependence relations among random variables, without the need to know the properties of individual variables as well as without the requirement to specify parametric family of entire underlying distribution for individual variables. Experiments on two real-application data sets show the effectiveness of the proposed method.     

The copula echo state network

Echo state networks (ESNs) constitute a novel approach to recurrent neural network (RNN) training, with an RNN (the reservoir) being generated randomly, and only a readout being trained using a simple, computationally efficient algorithm. ESNs have greatly facilitated the practical application of RNNs, outperforming classical approaches on a number of benchmark tasks. This paper studies the formulation of a class of copula-based semiparametric models for sequential data modeling, characterized by nonparametric marginal distributions modeled by postulating suitable echo state networks, and parametric copula functions that help capture all the scale-free temporal dependence of the modeled processes. We provide a simple algorithm for the data-driven estimation of the marginal distribution and the copula parameters of our model under the maximum-likelihood framework. We exhibit the merits of our approach by considering a number of applications; as we show, our method offers a significant enhancement in the dynamical data modeling capabilities of ESNs, without significant compromises in the algorithm's computational efficiency.     

Modeling of Asymmetry between Gasoline and Crude Oil Prices: A Monte Carlo Comparison

An Engle–Granger two-step procedure is commonly used to estimate cointegrating vectors and consequently asymmetric error-correction models. This study uses Monte Carlo methods and demonstrates that the Engle–Granger two-step method leads to biased estimates of asymmetric parameters and in some cases suggests symmetry in the asymmetric data generating process (DGP). The single equation error correction models (SEECM) based on ordinary least squares (OLS) and nonlinear least squares (NLS) are employed for simultaneous estimation of the cointegrating vector and the ECM. The SEECMs perform better than Engle–Granger two-step procedures in estimating the asymmetry and making inferences on its existence in various DGPs. We show that SEECM estimations are less biased and inferences are less likely to be misleading compared to the Engle–Granger two-step procedure. Unlike the asymmetric specifications based on Engle–Granger two-step approach, the asymmetric SEECMs do not refute the possibility of long-run asymmetry by allowing different cointegrating vectors for positive and negative regimes. Examining the model with real data also supports the Monte Carlo results. While the conventional approaches imply symmetry, the proposed asymmetric SEECM, which has been embedded in a Threshold Autoregressive model, uncovers asymmetry at the presence of different cointegrating vectors for positive and negative regimes.     

Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data

Penalized B-splines combined with the composite link model are used to estimate a bivariate density from a histogram with wide bins. The goals are multiple: they include the visualization of the dependence between the two variates, but also the estimation of derived quantities like Kendall’s tau, conditional moments and quantiles. Two strategies are proposed: the first one is semiparametric with flexible margins modeled using B-splines and a parametric copula for the dependence structure; the second one is nonparametric and is based on Kronecker products of the marginal B-spline bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of the two methods under different dependence structures and for varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz’s BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.     

Tail dependence structure of the foreign exchange market: A network view

Tail dependence of financial entities describes when the price of one financial asset has an extreme fluctuation (e.g., price sharply rises or falls), the degree of its effect on the price fluctuation of another asset. Under the background of the global financial crisis, tail dependence structure of financial entities plays an important role in financial risk management, portfolio selection, and asset pricing. In this paper, we propose a concept of tail dependence networks to investigate the tail dependence structure of the foreign exchange (FX) market. Lower- and upper-tail dependence networks for 42 major currencies in the FX market from 2005 to 2012 are constructed by combing the symmetrized Joe-Clayton copula model and two filtered graph algorithms, i.e., the minimum spanning tree (MST) and the planar maximally filtered graph (PMFG). We also construct the tail dependence hierarchical trees (HTs) associated with the MSTs to analyze the currency clusters. We find that (1) the two series of lower- and upper-tail dependence coefficients present different statistical properties; (2) the upper-tail dependence networks are tighter than the lower-tail dependence networks; and (3) different currency clusters, cliques and communities are respectively found in the two tail dependence networks. The key empirical results indicate that market participants should consider the different topological features at different market situations (e.g., a booming market or a recession market) to make decisions on the investing or hedging strategies. Overall, our obtained results based on the tail dependence networks are new insights in financial management and supply a novel analytical tool for market participants.     

Efficient estimation of a semiparametric dynamic copula model

A new semiparametric dynamic copula model is proposed where the marginals are specified as parametric GARCH-type processes, and the dependence parameter of the copula is allowed to change over time in a nonparametric way. A straightforward two-stage estimation method is given by local maximum likelihood for the dependence parameter, conditional on consistent first stage estimates of the marginals. First, the properties of the estimator are characterized in terms of bias and variance and the bandwidth selection problem is discussed. The proposed estimator attains the semiparametric efficiency bound and its superiority is demonstrated through simulations. Finally, the wide applicability of the model in financial time series is illustrated, and it is compared with traditional models based on conditional correlations.     

Comparison of semiparametric and parametric methods for estimating copulas

Copulas have attracted significant attention in the recent literature for modeling multivariate observations. An important feature of copulas is that they enable us to specify the univariate marginal distributions and their joint behavior separately. The copula parameter captures the intrinsic dependence between the marginal variables and it can be estimated by parametric or semiparametric methods. For practical applications, the so called inference function for margins (IFM) method has emerged as the preferred fully parametric method because it is close to maximum likelihood (ML) in approach and is easier to implement. The purpose of this paper is to compare the ML and IFM methods with a semiparametric (SP) method that treats the univariate marginal distributions as unknown functions. In this paper, we consider the SP method proposed by Genest et al. [1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543-552], which has attracted considerable interest in the literature. The results of an extensive simulation study reported here show that the ML/IFM methods are nonrobust against misspecification of the marginal distributions, and that the SP method performs better than the ML and IFM methods, overall. A data example on household expenditure is used to illustrate the application of various data analytic methods for applying the SP method, and to compare and contrast the ML, IFM and SP methods. The main conclusion is that, in terms of statistical computations and data analysis, the SP method is better than ML and IFM methods when the marginal distributions are unknown which is almost always the case in practice.