Modelling soccer matches using bivariate discrete distributions with general dependence structure

In this paper copulas are used to generate bivariate discrete distributions. These distributions are fitted to soccer data from the English Premier League. An interesting aspect of these data is that the primary variable of interest, the discrete pair shots-for and shots-against, exhibits negative dependence; thus, in particular, we apply bivariate Poisson-related distributions that allow such dependence. The paper focuses on Archimedian copulas, for which the dependence structure is fully determined by a one-dimensional projection that is invariant under marginal transformations. Diagnostic plots for copula fit based on this projection are adapted to deal with discrete variables. Covariates relating to within-match contributions such as numbers of passes and tackles are introduced to explain variability in shot outcomes. The results of this analysis would appear to support the notion that playing the 'beautiful game' is an effective strategy—more passes and crosses contribute to more effective play and more shots on the goal.     

Modelling dependence using skew t copulas: Bayesian inference and applications

We construct a copula from the skew t distribution of Sahu et al. ( 2003 ). This copula can capture asymmetric and extreme dependence between variables, and is one of the few copulas that can do so and still be used in high dimensions effectively. However, it is difficult to estimate the copula model by maximum likelihood when the multivariate dimension is high, or when some or all of the marginal distributions are discrete‐valued, or when the parameters in the marginal distributions and copula are estimated jointly. We therefore propose a Bayesian approach that overcomes all these problems. The computations are undertaken using a Markov chain Monte Carlo simulation method which exploits the conditionally Gaussian representation of the skew t distribution. We employ the approach in two contemporary econometric studies. The first is the modelling of regional spot prices in the Australian electricity market. Here, we observe complex non‐Gaussian margins and nonlinear inter‐regional dependence. Accurate characterization of this dependence is important for the study of market integration and risk management purposes. The second is the modelling of ordinal exposure measures for 15 major websites. Dependence between websites is important when measuring the impact of multi‐site advertising campaigns. In both cases the skew t copula substantially outperforms symmetric elliptical copula alternatives, demonstrating that the skew t copula is a powerful modelling tool when coupled with Bayesian inference. Copyright © 2010 John Wiley & Sons, Ltd.     

Improving forecasting performance using covariate-dependent copula models

Copulas provide an attractive approach to the construction of multivariate distributions with flexible marginal distributions and different forms of dependences. Of particular importance in many areas is the possibility of forecasting the tail-dependences explicitly. Most of the available approaches are only able to estimate tail-dependences and correlations via nuisance parameters, and cannot be used for either interpretation or forecasting. We propose a general Bayesian approach for modeling and forecasting tail-dependences and correlations as explicit functions of covariates, with the aim of improving the copula forecasting performance. The proposed covariate-dependent copula model also allows for Bayesian variable selection from among the covariates of the marginal models, as well as the copula density. The copulas that we study include the Joe-Clayton copula, the Clayton copula, the Gumbel copula and the Student’s $t$-copula. Posterior inference is carried out using an efficient MCMC simulation method. Our approach is applied to both simulated data and the S&P 100 and S&P 600 stock indices. The forecasting performance of the proposed approach is compared with those of other modeling strategies based on log predictive scores. A value-at-risk evaluation is also performed for the model comparisons.     

Empirical evidence of extreme dependence and contagion risk between main cryptocurrencies

This study examines the dependence and contagion risk between Bitcoin (BTC), Litecoin (LTC) and Ripple (XRP) using non-parametric mixture copulas (developed by Zimmer, 2012) and recently proposed methods of full-range tail dependence copulas (advanced by Hua, 2017, Su and Hua, 2017), for the period from 04-08-2013 to 17-06-2018. The Chi-plots and Kendall plots results show heavy tail dependence between each pairs of the cryptocurrencies. Evidence from the mixture copula indicates that for the BTC-LTC pair the upper-tail dependence is both stronger and more prevalent, while for the other pairs of cryptocurrencies the lower-tail dependence is very strong and more prevalent. However, the results of the full-range tail dependence copulas reveal a strong and prevalent upper and lower-tail dependence of each pairs of cryptocurrencies. These results provide evidence of significant risk contagion among price returns of major cryptocurrencies, both in bull and bear markets.     

Constructing Generalized FGM Copulas by Means of Certain Univariate Distributions

In this paper we focus on specific generalized Fairlie- Gumbel-Morgenstern (or Sarmanov) copulas which are generated by a single function (so-called generator or generator function) defined on the unit interval. In particular, we introduce a class of generators based on density-quantile functions of certain univariate distributions. Many of the generator functions from the literature are recovered as special cases. Moreover, two new generators are suggested, implying to new copulas. Finally, the opposite way around, it is shown how to calculate the univariate distribution which belongs to a given copula generator function.     

Time series copulas for heteroskedastic data

We propose parametric copulas that capture serial dependence in stationary heteroskedastic time series. We suggest copulas for first‐order Markov series, and then extend them to higher orders and multivariate series. We derive the copula of a volatility proxy, based on which we propose new measures of volatility dependence, including co‐movement and spillover in multivariate series. In general, these depend upon the marginal distributions of the series. Using exchange rate returns, we show that the resulting copula models can capture their marginal distributions more accurately than univariate and multivariate generalized autoregressive conditional heteroskedasticity models, and produce more accurate value‐at‐risk forecasts.     

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.      

A method to obtain new copulas from a given one

Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ^[−1], conditions that φ must satisfy for C_φ(x₁, . . . ,x_n)=φ^[−1](C(φ(x₁), . . . ,φ(x_n))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and several examples of generator functions φ that can be used to construct copulas C_φ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related with mixtures     

Financial dependence analysis: applications of vine copulas

This paper features the application of a novel and recently developed method of statistical and mathematical analysis to the assessment of financial risk, namely regular vine copulas. Dependence modelling using copulas is a popular tool in financial applications but is usually applied to pairs of securities. Vine copulas offer greater flexibility and permit the modelling of complex dependence patterns using the rich variety of bivariate copulas that can be arranged and analysed in a tree structure to facilitate the analysis of multiple dependencies. We apply regular vine copula analysis to a sample of stocks comprising the Dow Jones index to assess their interdependencies and to assess how their correlations change in different economic circumstances using three different sample periods around Global Financial Crisis (GFC).: pre‐GFC (January 2005 to July 2007), GFC (July 2007 to September 2009) and post‐GFC periods (September 2009 to December 2011). The empirical results suggest that the dependencies change in a complex manner, and there is evidence of greater reliance on the Student‐t copula in the copula choice within the tree structures for the GFC period, which is consistent with the existence of larger tails in the distributions of returns for this period. One of the attractions of this approach to risk modelling is the flexibility in the choice of distributions used to model co‐dependencies. The practical application of regular vine metrics is demonstrated via an example of the calculation of the Value at Risk of a portfolio of stocks.     

Predicting the portfolio risk of high-dimensional international stock indices with dynamic spatial dependence

The spatial dependence of assets, which relates to similarities in economic, political, or cultural systems and other aspects, has been confirmed through empirical research; however, spatial dependence has rarely been applied to financial risk measurement. To fill this gap in the literature, a dynamic spatial GARCH-copula (sGC) model is proposed in this paper to evaluate the portfolio risk of international stock indices. In this model, a spatial GARCH is used as the marginal distribution and vine copula is adopted as the joint distribution of indices. Then, the proposed model is applied empirically to assess portfolio risk. Results show that, first, the proposed risk prediction model with spatial dependence outperforms a model neglecting spatial effects per the Kupiec test, Z test and Christoffersen test. Risk prediction during periods of economic stability is also more accurate than during times of crisis. Second, risk measures for models with spatial dependence are higher than those without such dependence but lower than for vine copula models. Third, models including either spatial dependence or vine copulas alone exhibit relatively poor performance. Fourth, the model involving extreme value theory (EVT) generates the greatest value at risk to pass the Kupiec test, Z test and Christoffersen test; however, this model is not suitable for characterizing international indices with EVT based on negative values of the shape parameters of estimates. Findings offer important implications for personal investors, institutional investors, and national regulatory authorities.     

Estimation of copula-based semiparametric time series models

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric marginal distributions and parametric copula functions, while the copulas capture all the scale-free temporal dependence of the processes. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. These results are used to obtain root-n consistent and asymptotically normal estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles. The semiparametric conditional quantile estimators are automatically monotonic across quantiles, which is attractive for portfolio conditional value-at-risk calculations.     

Estimation and model selection of semiparametric multivariate survival functions under general censorship

We study estimation and model selection of semiparametric models of multivariate survival functions for censored data, which are characterized by possibly misspecified parametric copulas and nonparametric marginal survivals. We obtain the consistency and root-n$n$ asymptotic normality of a two-step copula estimator to the pseudo-true copula parameter value according to KLIC, and provide a simple consistent estimator of its asymptotic variance, allowing for a first-step nonparametric estimation of the marginal survivals. We establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application is provided.     

Flexible Estimation of Copulas: An Application to the US Housing Crisis

Zimmer (‘The role of copulas in the housing crisis’, Review of Economics and Statistics 2012; 94 : 607–620) provides an interesting case study of the pitfalls of using parametric copulas to understand the US housing crisis in the latter part of 2000s. The original study by Zimmer (2012) employs a finite‐mixture copula to illustrate that the symmetry of the Gaussian copula may not be tenable, especially for US housing price data during the time period from 1975:Q2 to 2009:Q1. We undertake a replication of his study in a wide sense. First, we replicate the study by incorporating revised data and then extending the dataset to include the most recent data. Second, we implement a nonparametric copula estimator recently proposed by Racine (‘Mixed data kernel copulas’, Empirical Economics forthcoming) to the parametrically filtered data used in Zimmer (2012). Our replication finds that the application of the nonparametric copula to the same and extended filtered data provides an alternative flexible specification for copulas. However, the overall cautionary message of the flexible‐form copula espoused in Zimmer (2012) remains. Copyright © 2015 John Wiley & Sons, Ltd.     

Diversification for general copula dependence

We generalize the extreme value analysis for Archimedean copulas (see Alink , Löwe and Wüthrich , 2003) to the non-Archimedean case: Assume we have d ≥2 exchangeable and continuously distributed risks X ₁,…, X _d . Under appropriate assumptions there is a constant q _d such that, for all large u , we have . The constant q _d describes the asymptotic dependence structure. Typically, q _d will depend on more aspects of this dependence structure than the well-known tail dependence coefficient.     

Switching generalized autoregressive score copula models with application to systemic risk

Recent financial disasters have emphasized the need to accurately predict extreme financial losses and their consequences for the institutions belonging to a given financial market. The ability of econometric models to predict extreme events strongly relies on their flexibility to account for the highly nonlinear and asymmetric dependence patterns observed in financial time series. In this paper, we develop a new class of flexible copula models where the dependence parameters evolve according to a Markov switching generalized autoregressive score (GAS) dynamics. Maximum likelihood estimation is performed using a two‐step procedure where the second step relies on the expectation–maximization algorithm. The proposed switching GAS copula models are then used to estimate the conditional value at risk and the conditional expected shortfall, measuring the impact on an institution of extreme events affecting another institution or the market. The empirical investigation, conducted on a panel of European regional portfolios, reveals that the proposed model is able to explain and predict the evolution of the systemic risk contributions over the period 1999–2015.     

Relationship between oil,stock prices and exchange rates: A vine copula based GARCH method

In this paper, we apply a vine copula approach to investigate the dynamic relationship between energy, stock and currency markets. Dependence modeling using vine copulas offers a greater flexibility and permits the modeling of complex dependency patterns for high-dimensional distributions. Using a sample of more than 10 years of daily return observations of the WTI crude oil, the Dow Jones Industrial average stock index and the trade weighted US dollar index returns, we find evidence of a significant and symmetric relationship between these variables. Considering different sample periods show that the dynamic of the relationship between returns is not constant over time. Our results indicate also that the dependence structure is highly affected by the financial crisis and Great Recession, over 2007–2009. Finally, there is evidence to suggest that the application of the vine copula model improves the accuracy of VaR estimates, compared to traditional approaches.     

Comparing the accuracy of multivariate density forecasts in selected regions of the copula support

This paper develops a testing framework for comparing the predictive accuracy of competing multivariate density forecasts with different predictive copulas, focusing on specific parts of the copula support. The tests are framed in the context of the Kullback–Leibler Information Criterion, using (out-of-sample) conditional likelihood and censored likelihood in order to focus the evaluation on the region of interest. Monte Carlo simulations document that the resulting test statistics have satisfactory size and power properties for realistic sample sizes. In an empirical application to daily changes of yields on government bonds of the G7 countries we obtain insights into why the Student-t and Clayton mixture copula outperforms the other copulas considered; mixing in the Clayton copula with the t-copula is of particular importance to obtain high forecast accuracy in periods of jointly falling yields.     

Stochastic comparisons of multivariate reversed frailty model

Multivariate frailty approaches are most commonly used to define distributions of random vectors, which represent lifetimes of individuals or components and stochastically compare them in terms of various multivariate orders. In this paper, we study a multivariate shared reversed frailty model and a general multivariate reversed frailty mixture model, and derive sufficient conditions for some of the stochastic orderings to hold among the random vectors. We also consider a particular case of a general multivariate mixture model in which the baseline distribution function is represented in terms of a copula and study stochastic comparisons (stochastic and lower orthant order) among the two random vectors.     

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.     

Likelihood-based estimation in a panel setting: Robustness,redundancy and validity of copulas

This paper considers the estimation of likelihood-based models in a panel setting. That is, we have panel data, and for each time period separately we have a correctly specified model that could be estimated by MLE. We want to allow non-independence over time. This paper shows how to improve on the QMLE. It then considers MLE based on joint distributions constructed using copulas. It discusses the efficiency gain from using the true copula, and shows that knowledge of the true copula is redundant only if the variance matrix of the relevant set of moment conditions is singular. It also discusses the question of robustness against misspecification of the copula, and proposes a test of the validity of the copula. GMM methods are argued to be useful analytically, and also for reasons of efficiency if the copula is robust but not correct.