Pair-copula Construction Research Articles

On the weak convergence of the empirical conditional copula under a simplifying assumption

A common assumption in pair-copula constructions is that the copula of the conditional distribution of two random variables given a covariate does not depend on the value of that covariate. Two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. On the one hand, the best possible rates for estimating the marginal conditional distribution functions are slower than the parametric one. On the other hand, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by Hobæk Haff and Segers (2015) and improving upon the nonparametric rate obtained theoretically by Gijbels et al. (2015). The novelty of the proposed approach lies in a double smoothing procedure for the estimator of the marginal conditional distribution functions. The copula estimator itself is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional distribution functions were known.

Application of selection and estimation regular vine copula on go public company share

The accuracy of financial risk management involving a large number of assets is needed, but information about dependencies among assets cannot be adequately analyzed. To analyze dependencies on a number of assets, several tools have been added to standard multivariate copula. However, these tools have not been adequately used in apps with higher dimensions. The bivariate parametric copula families can be used to solve it. The multivariate copula can be built from the bivariate parametric copula which is connected by a graphical representation to become Pair Copula Constructions (PCCs) or vine copula. The application of C-vine and D-vine copula have been used in some researches, but the use of C-vine and D-vine copula is more limited than R-vine copula. Therefore, this study used R-vine copula to provide flexibility for modeling complex dependencies on a high dimension. Since copula is a static model, while stock values change over time, then copula should be combined with the ARMA- GARCH model for modeling the movement of shares (volatility). The objective of this paper is to select and estimate R-vine copula which is used to analyze PT Jasa Marga (Persero) Tbk (JSMR), PT Waskita Karya (Persero) Tbk (WSKT), and PT Bank Mandiri (Persero) Tbk (BMRI) from august 31, 2014 to august 31, 2017. From the method it is obtained that the selected copulas for 2 edges at the first tree are survival Gumbel and the copula for edge at the second tree is Gaussian.

Joint regression modeling for missing categorical covariates in generalized linear models

Missing covariates data is a common issue in generalized linear models (GLMs). A model-based procedure arising from properly specifying joint models for both the partially observed covariates and the corresponding missing indicator variables represents a sound and flexible methodology, which lends itself to maximum likelihood estimation as the likelihood function is available in computable form. In this paper, a novel model-based methodology is proposed for the regression analysis of GLMs when the partially observed covariates are categorical. Pair-copula constructions are used as graphical tools in order to facilitate the specification of the high-dimensional probability distributions of the underlying missingness components. The model parameters are estimated by maximizing the weighted log-likelihood function by using an EM algorithm. In order to compare the performance of the proposed methodology with other well-established approaches, which include complete-cases and multiple imputation, several simulation experiments of Binomial, Poisson and Normal regressions are carried out under both missing at random and non-missing at random mechanisms scenarios. The methods are illustrated by modeling data from a stage III melanoma clinical trial. The results show that the methodology is rather robust and flexible, representing a competitive alternative to traditional techniques.

Pair Copula Constructions for Insurance Experience Rating

ABSTRACTIn nonlife insurance, insurers use experience rating to adjust premiums to reflect policyholders’ previous claim experience. Performing prospective experience rating can be challenging when the claim distribution is complex. For instance, insurance claims are semicontinuous in that a fraction of zeros is often associated with an otherwise positive continuous outcome from a right-skewed and long-tailed distribution. Practitioners use credibility premium that is a special form of the shrinkage estimator in the longitudinal data framework. However, the linear predictor is not informative especially when the outcome follows a mixed distribution. In this article, we introduce a mixed vine pair copula construction framework for modeling semicontinuous longitudinal claims. In the proposed framework, a two-component mixture regression is employed to accommodate the zero inflation and thick tails in the claim distribution. The temporal dependence among repeated observations is modeled using a sequence of bivariate conditional copulas based on a mixed D-vine. We emphasize that the resulting predictive distribution allows insurers to incorporate past experience into future premiums in a nonlinear fashion and the classic linear predictor can be viewed as a nested case. In the application, we examine a unique claims dataset of government property insurance from the state of Wisconsin. Due to the discrepancies between the claim and premium distributions, we employ an ordered Lorenz curve to evaluate the predictive performance. We show that the proposed approach offers substantial opportunities for separating risks and identifying profitable business when compared with alternative experience rating methods. Supplementary materials for this article are available online.

Dependent defaults and losses with factor copula models

Abstract We present a class of flexible and tractable static factor models for the term structure of joint default probabilities, the factor copula models. These high-dimensional models remain parsimonious with paircopula constructions, and nest many standard models as special cases. The loss distribution of a portfolio of contingent claims can be exactly and efficiently computed when individual losses are discretely supported on a finite grid. Numerical examples study the key features affecting the loss distribution and multi-name credit derivatives prices. An empirical exercise illustrates the flexibility of our approach by fitting credit index tranche prices.

Approximating non-Gaussian Bayesian networks using minimum information vine model with applications in financial modelling

Many financial modeling applications require to jointly model multiple uncertain quantities to present more accurate, near future probabilistic predictions. Informed decision making would certainly benefit from such predictions. Bayesian networks (BNs) and copulas are widely used for modeling numerous uncertain scenarios. Copulas, in particular, have attracted more interest due to their nice property of approximating the probability distribution of the data with heavy tail. Heavy tail data is frequently observed in financial applications. The standard multivariate copula suffer from serious limitations which made them unsuitable for modeling the financial data. An alternative copula model called the pair-copula construction (PCC) model is more flexible and efficient for modeling the complex dependence of financial data. The only restriction of PCC model is the challenge of selecting the best model structure. This issue can be tackled by capturing conditional independence using the Bayesian network PCC (BN-PCC). The flexible structure of this model can be derived from conditional independences statements learned from data. Additionally, the difficulty of computing conditional distributions in graphical models for non-Gaussian distributions can be eased using pair-copulas. In this paper, we extend this approach further using the minimum information vine model which results in a more flexible and efficient approach in understanding the complex dependence between multiple variables with heavy tail dependence and asymmetric features which appear widely in the financial applications.

Recent Developments in Copula Models

Copula models have become very popular and well studied among the scientific community.[...]

Multivariate statistical modelling of compound events via pair-copula constructions: analysis of floods in Ravenna (Italy)

Abstract. Compound events (CEs) are multivariate extreme events in which the individual contributing variables may not be extreme themselves, but their joint – dependent – occurrence causes an extreme impact. Conventional univariate statistical analysis cannot give accurate information regarding the multivariate nature of these events. We develop a conceptual model, implemented via pair-copula constructions, which allows for the quantification of the risk associated with compound events in present-day and future climate, as well as the uncertainty estimates around such risk. The model includes predictors, which could represent for instance meteorological processes that provide insight into both the involved physical mechanisms and the temporal variability of compound events. Moreover, this model enables multivariate statistical downscaling of compound events. Downscaling is required to extend the compound events' risk assessment to the past or future climate, where climate models either do not simulate realistic values of the local variables driving the events or do not simulate them at all. Based on the developed model, we study compound floods, i.e. joint storm surge and high river runoff, in Ravenna (Italy). To explicitly quantify the risk, we define the impact of compound floods as a function of sea and river levels. We use meteorological predictors to extend the analysis to the past, and get a more robust risk analysis. We quantify the uncertainties of the risk analysis, observing that they are very large due to the shortness of the available data, though this may also be the case in other studies where they have not been estimated. Ignoring the dependence between sea and river levels would result in an underestimation of risk; in particular, the expected return period of the highest compound flood observed increases from about 20 to 32 years when switching from the dependent to the independent case.

Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting

Factor modeling is a popular strategy to induce sparsity in multivariate models as they scale to higher dimensions. We develop Bayesian inference for a recently proposed latent factor copula model, which utilizes a pair copula construction to couple the variables with the latent factor. We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling for posterior simulation: Gibbs sampling enables application to Bayesian problems, while ARMS is an adaptive strategy that replaces traditional Metropolis-Hastings updates, which typically require careful tuning. Our simulation study shows favorable performance of our proposed approach both in terms of sampling efficiency and accuracy. We provide an extensive application example using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES).

Construction and identification of a D-Vine model applied to the probability distribution of modal parameters in structural dynamics

This study investigates the construction and identification of the probability distribution of random modal parameters (natural frequencies and effective parameters) in structural dynamics. As these parameters present various types of dependence structures, the retained approach is based on pair copula construction (PCC). A literature review leads us to choose a D-Vine model for the construction of modal parameters probability distributions. Identification of this model is based on likelihood maximization which makes it sensitive to the dimension of the distribution, namely the number of considered modes in our context. To this respect, a mode selection preprocessing step is proposed. It allows the selection of the relevant random modes for a given transfer function. The second point, addressed in this study, concerns the choice of the D-Vine model. Indeed, D-Vine model is not uniquely defined. Two strategies are proposed and compared. The first one is based on the context of the study whereas the second one is purely based on statistical considerations. Finally, the proposed approaches are numerically studied and compared with respect to their capabilities, first in the identification of the probability distribution of random modal parameters and second in the estimation of the 99% quantiles of some transfer functions.

Modeling Influenza-Like Illness Activity in the United States

Influenza causes yearly costs for hospitalization and outpatient visits of more than $10 billion in the United States. The prediction of influenza epidemics is thus relevant for health insurance providers and public health facilities, among others. A useful piece of information is the probability distribution of influenza epidemics occurring within a given time horizon of one or two years. We present a model that delivers confidence intervals for future influenza activity in different regions in the United States. The model takes into account the specific statistical characteristics of influenza activity such as volatility clusters, seasonal effects, and dependencies between different regions. Confidence intervals for the regions are obtained using ARMA-GARCH models, and regional dependencies are captured by a pair-copula construction, describing jointly the residuals of the ARMA-GARCH models. Our model allows us to simulate influenza activity over a future time horizon.

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 non-parametric 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.

Vine copula approximation: a generic method for coping with conditional dependence

Pair-copula constructions (or vine copulas) are structured, in the layout of vines, with bivariate copulas and conditional bivariate copulas. The main contribution of the current work is an approach to the long-standing problem: how to cope with the dependence structure between the two conditioned variables indicated by an edge, acknowledging that the dependence structure changes with the values of the conditioning variables. The changeable dependence problem, though recognized as crucial in the field of multivariate modelling, remains widely unexplored due to its inherent complication and hence is the motivation of the current work. Rather than resorting to traditional parametric or nonparametric methods, we proceed from an innovative viewpoint: approximating a conditional copula, to any required degree of approximation, by utilizing a family of basis functions. We fully incorporate the impact of the conditioning variables on the functional form of a conditional copula by employing local learning methods. The attractions and dilemmas of the pair-copula approximating technique are revealed via simulated data, and its practical importance is evidenced via a real data set.

D-vine copula based quantile regression

Quantile regression, that is the prediction of conditional quantiles, has steadily gained importance in statistical modeling and financial applications. A new semiparametric quantile regression method is introduced. It is based on sequentially fitting a likelihood optimal D-vine copula to given data resulting in highly flexible models with easily extractable conditional quantiles. As a subclass of regular vine copulas, D-vines enable the modeling of multivariate copulas in terms of bivariate building blocks, a so-called pair-copula construction (PCC). The proposed algorithm works fast and accurate even in high dimensions and incorporates an automatic variable selection by maximizing the conditional log-likelihood. Further, typical issues of quantile regression such as quantile crossing or transformations, interactions and collinearity of variables are automatically taken care of. In a simulation study the improved accuracy and reduced computation time of the approach in comparison with established quantile regression methods is highlighted. An extensive financial application to international credit default swap (CDS) data including stress testing and Value-at-Risk (VaR) prediction demonstrates the usefulness of the proposed method.

Pair-Copula Constructions for Financial Applications: A Review

This survey reviews the large and growing literature on the use of pair-copula constructions (PCCs) in financial applications. Using a PCC, multivariate data that exhibit complex patterns of dependence can be modeled using bivariate copulae as simple building blocks. Hence, this model represents a very flexible way of constructing higher-dimensional copulae. In this paper, we survey inference methods and goodness-of-fit tests for such models, as well as empirical applications of the PCCs in finance and economics.

A goodness-of-fit test for regular vine copula models

ABSTRACTWe introduce a new goodness-of-fit test for regular vine (R-vine) copula models, a very flexible class of multivariate copulas based on a pair-copula construction (PCC). The test arises from White’s information matrix test and extends an existing goodness-of-fit test for copulas. The corresponding critical value can be approximated by asymptotic theory or simulation. The simulation based test shows excellent performance with regard to observed size and power in an extensive simulation study, while the asymptotic theory based test is inadequate for n≤10,000 for a 5-dimensional model (in d = 8 even 20,000 are not enough). The simulation based test is applied to select among different R-vine specifications modeling the dependency among exchange rates.

Pair-Copula Bayesian Networks

Pair-copula Bayesian networks (PCBNs) are a novel class of multivariate statistical models, which combine the distributional flexibility of pair-copula constructions (PCCs) with the parsimony of conditional independence models associated with directed acyclic graphs (DAGs). We are first to provide generic algorithms for random sampling and likelihood inference in arbitrary PCBNs as well as for selecting orderings of the parents of the vertices in the underlying graphs. Model selection of the DAG is facilitated using a version of the well-known PC algorithm that is based on a novel test for conditional independence of random variables tailored to the PCC framework. A simulation study shows the PC algorithm’s high aptitude for structure estimation in non-Gaussian PCBNs. The proposed methods are finally applied to modeling financial return data. Supplementary materials for this article are available online.

Generalized Additive Models for Pair-Copula Constructions

Pair-copula constructions are flexible models for the dependence in a random vector that have attracted a lot of interest in recent years. In this paper, we use generalized additive models to extend pair-copula constructions to allow for effects of covariates on the dependence parameters. We let each pair-copula parameter depend directly on the covariates in a parametric, semi-parametric or non-parametric way. We propose a sequential estimation method that we study by simulation, and apply our method to investigate the time-varying dependence structure between the intraday returns on four major foreign exchange rates.

Probabilistic modeling of flood characterizations with parametric and minimum information pair-copula model

This paper highlights the usefulness of the minimum information and parametric pair-copula construction (PCC) to model the joint distribution of flood event properties. Both of these models outperform other standard multivariate copula in modeling multivariate flood data that exhibiting complex patterns of dependence, particularly in the tails. In particular, the minimum information pair-copula model shows greater flexibility and produces better approximation of the joint probability density and corresponding measures have capability for effective hazard assessments. The study demonstrates that any multivariate density can be approximated to any degree of desired precision using minimum information pair-copula model and can be practically used for probabilistic flood hazard assessment.

Modeling high‐dimensional time‐varying dependence using dynamic D‐vine models

We consider the problem of modeling the dependence among many time series. We build high‐dimensional time‐varying copula models by combining pair‐copula constructions with stochastic autoregressive copula and generalized autoregressive score models to capture dependence that changes over time. We show how the estimation of this highly complex model can be broken down into the estimation of a sequence of bivariate models, which can be achieved by using the method of maximum likelihood. Further, by restricting the conditional dependence parameter on higher cascades of the pair copula construction to be constant, we can greatly reduce the number of parameters to be estimated without losing much flexibility. Applications to five MSCI stock market indices and to a large dataset of daily stock returns of all constituents of the Dax 30 illustrate the usefulness of the proposed model class in‐sample and for density forecasting. Copyright © 2016 John Wiley & Sons, Ltd.