Conditional Copula Research Articles

Nonparametric estimation of multivariate multiparameter conditional copulas

Nonparametric estimation of conditional copulas with one parameter has been investigated in Acar et al. (2011). The estimation for multivariate multiparameter conditional copulas, however, has not been considered so far. This paper adopts the local linear smoothing technique and Newton–Raphson method to estimate those copulas. Under some regularity conditions, the asymptotic normality of the estimators is obtained. Simulation work shows the efficiency of the proposed method. As an application, we analyze a life expectancies data set and show that the conditional t copula outperforms the conditional Clayton, Frank and Gumbel copulas.

A development of downscaling scheme for sub-daily extreme precipitation using conditional copula model

A development of downscaling scheme for sub-daily extreme precipitation using conditional copula model

Dynamic conditional copula correlation and optimal hedge ratios with currency futures

This study investigates efficiency of the futures hedge implemented through the currency markets. The copula DCC-EGARCH model is estimated with the bivariate error correction term to minimize variance of the currency portfolios. The estimation results for the currencies of the Australian dollar, Canadian dollar, euro, British pound and Japanese yen show that the inclusion of the external realized variance estimators into the variance equation of the estimated model improves the model's ability to account for the clustered data variance. In hedging portfolios, the information content of the realized variance estimator effectively reduces the variance of the portfolios.

Vine copula models with GLM and sparsity

ABSTRACTVine copula provides a flexible tool to capture asymmetry in modeling multivariate distributions. Nevertheless, its flexibility is achieved at the expense of exponentially increasing complexity of the model. To alleviate this issue, the simplifying assumption (SA) is commonly adapted in specific applications of vine copula models. In this paper, generalized linear models (GLMs) are proposed for the parameters in conditional bivariate copulas to relax the SA. In the spirit of the principle of parsimony, a regularization methodology is developed to control the number of parameters, leading to sparse vine copula models. The conventional vine copula with the SA, the proposed GLM-based vine copula, and the sparse vine copula are applied to several financial datasets, and the results show that our proposed models outperform the one with SA significantly in terms of the Bayesian information criterion.

Conditional Correlation Coefficient as a Tool for Analysis of Contagion in Financial Markets and Real Economy Indexes Based on the Synthetic Ratio

We define contagion in financial markets as a significant increase in cross-market linkages after a shock to one or group of countries. Contagion occurs if cross-market co-movement increases significantly after the shock. In this article, the authors attempt to answer the question whether the selected world stock exchanges and economies are infecting each other within the meaning of the definition provided. Conditional copula functions and conditional Spearman's correlation coefficient will be used as a tool. Construction of a synthetic index of world financial markets is introduced. This index is based on the taxonomic distances of chosen stock market from the “best” object, where the best means object which has highest financial parameters.Main goal of this paper is to analyze changes in dependence between US stock market (S&P500) and chosen groups of world stock markets and constructed real economy index.

Sparse conditional copula models for structured output regression

We deal with the multiple output regression task where the central theme is to capture the sparse output correlation among the output variables. Sparse inverse covariance learning of linear Gaussian conditional models has been recently studied, shown to achieve superb prediction performance. However, it can fail when the underlying true input–output process is non-Gaussian and/or non-linear. We introduce a novel sparse conditional copula model to represent the joint density of the output variables. By incorporating a Gaussian copula function, yet modeling univariate marginal densities by (non-Gaussian) mixtures of experts, we achieve high flexibility in representation that admits non-linear and non-Gaussian densities. We then propose a sparse learning method for this copula-based model that effectively imposes sparsity in the conditional dependency among output variables. The learning optimization is efficient as it can be decomposed into gradient-descent marginal density estimation and the sparse inverse covariance learning for the copula function. Improved performance of the proposed approach is demonstrated on several interesting image/vision tasks with high dimensions.

Time‐varying nonstationary multivariate risk analysis using a dynamic Bayesian copula

AbstractA time‐varying risk analysis is proposed for an adaptive design framework in nonstationary conditions arising from climate change. A Bayesian, dynamic conditional copula is developed for modeling the time‐varying dependence structure between mixed continuous and discrete multiattributes of multidimensional hydrometeorological phenomena. Joint Bayesian inference is carried out to fit the marginals and copula in an illustrative example using an adaptive, Gibbs Markov Chain Monte Carlo (MCMC) sampler. Posterior mean estimates and credible intervals are provided for the model parameters and the Deviance Information Criterion (DIC) is used to select the model that best captures different forms of nonstationarity over time. This study also introduces a fully Bayesian, time‐varying joint return period for multivariate time‐dependent risk analysis in nonstationary environments. The results demonstrate that the nature and the risk of extreme‐climate multidimensional processes are changed over time under the impact of climate change, and accordingly the long‐term decision making strategies should be updated based on the anomalies of the nonstationary environment.

Properties of the probabilistic implications and S-implications

Recently, Grzegorzewski (2011) introduced two new families of fuzzy implication functions called probabilistic implications and probabilistic S-implications. They are based on conditional copulas and make a bridge between probability theory and fuzzy logic. In his previous articles author gave a motivation to his idea and indicates some interesting connections between new families of multivalued implications and the dependence structure of the underlying environment. In this paper the laws of contraposition, the law of importation and T-conditionality are studied for these families of fuzzy implications. Furthermore, we discuss the intersections of both new families of implications with R-implications, (S,N)-implications and QL-operations.

A stochastic power curve for wind turbines with reduced variability using conditional copula

AbstractIt has been observed that a large variability exists between wind speed and wind power in real metrological conditions. To reduce this substantial variability, this study developed a stochastic wind turbine power curve by incorporating various exogenous factors. Four measurements, namely, wind azimuth, wind elevation, air density and solar radiation are chosen as exogenous influence factors. A recursive formula based on conditional copulas is used to capture the complex dependency structure between wind speed and wind power with reduced variability. A procedure of selecting a proper form for each factor and its corresponding copula models is given. Through a case study on the small wind turbine located in southeast of Edmonton, Alberta, Canada, we demonstrate that the variability can be reduced significantly by incorporating these influence factors. Wind turbine operators can apply the method reported in this study to construct a stochastic power curve for local wind farms and use it to achieve more accurate power forecasting and health condition monitoring of the turbine. Copyright © 2015 John Wiley & Sons, Ltd.

Patched approximations and their convergence

Patched approximations of copulas unify ordinal sums, shuffles of Min, checkerboard, and checkmin approximations. We give a characterization of patched approximations and an error bound of the approximations in Sobolev norm. Patched approximations with uniform marginal conditional distributions are shown to arise naturally. We prove that these uniform patched approximations converge uniformly and in the Sobolev norm. The latter convergence is settled by showing the convergence almost everywhere of the first partial derivatives. We also show that the independence copula can be approximated by conditional mutual complete copulas in the Sobolev norm.

Copula-based method for multisite monthly and daily streamflow simulation

Multisite stochastic simulation of streamflow sequences is needed for water resources planning and management. In this study, a new copula-based method is proposed for generating long-term multisite monthly and daily streamflow data. A multivariate copula, which is established based on bivariate copulas and conditional probability distributions, is employed to describe temporal dependences (single site) and spatial dependences (between sites). Monthly or daily streamflows at multiple sites are then generated by sampling from the conditional copula. Three tributaries of the Colorado River and the upper Yangtze River are selected to evaluate the proposed methodology. Results show that the generated data at both longer and shorter time scales can capture the distribution properties of a single site and preserve the spatial correlation of streamflow at different locations. The main advantage of the method is that the trivariate copula can be established using three bivariate copulas and the model parameters can be easily estimated by the Kendall tau rank correlation coefficient. The method provides a new tool for multisite flow data stochastic simulation.

Simultaneous inference in structured additive conditional copula regression models: a unifying Bayesian approach

While most regression models focus on explaining distributional aspects of one single response variable alone, interest in modern statistical applications has recently shifted towards simultaneously studying multiple response variables as well as their dependence structure. A particularly useful tool for pursuing such an analysis are copula-based regression models since they enable the separation of the marginal response distributions and the dependence structure summarised in a specific copula model. However, so far copula-based regression models have mostly been relying on two-step approaches where the marginal distributions are determined first whereas the copula structure is studied in a second step after plugging in the estimated marginal distributions. Moreover, the parameters of the copula are mostly treated as a constant not related to covariates and most regression specifications for the marginals are restricted to purely linear predictors. We therefore propose simultaneous Bayesian inference for both the marginal distributions and the copula using computationally efficient Markov chain Monte Carlo simulation techniques. In addition, we replace the commonly used linear predictor by a generic structured additive predictor comprising for example nonlinear effects of continuous covariates, spatial effects or random effects and furthermore allow to make the copula parameters covariate-dependent. To facilitate Bayesian inference, we construct proposal densities for a Metropolis---Hastings algorithm relying on quadratic approximations to the full conditionals of regression coefficients avoiding manual tuning. The performance of the resulting Bayesian estimates is evaluated in simulations comparing our approach with penalised likelihood inference, studying the choice of a specific copula model based on the deviance information criterion, and comparing a simultaneous approach with a two-step procedure. Furthermore, the flexibility of Bayesian conditional copula regression models is illustrated in two applications on childhood undernutrition and macroecology.

Estimation of a Copula when a Covariate Affects only Marginal Distributions

AbstractThis paper is concerned with studying the dependence structure between two random variables Y1 and Y2 in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y1 and Y2 given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair‐copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location‐scale models for the effect of the covariate X on the marginals of Y1 and Y2 and show that in these cases, weak convergence of the estimator, at ‐rate, holds. The theoretical results are illustrated by simulations and a real data example.

Modeling the Dependence Structure between Australian Equity and Real Estate Markets – A Copula Approach

In the last decade, the Australian market for Real Estate Investment Trusts (REITS) has shown substantial growth rates. Australian Real Estate Investment Trusts (AREITS) are a unitized portfolio of property assets which allows investors to purchase a share in a diversified and professionally managed portfolio of real estate. We apply conditional copula models in order to investigate the dependence structure between returns of AREITS and Australian equity markets. The dynamic analysis is based on an AR-GARCH approach to model time-varying volatility of the individual series in combination with different copulas to capture the dependence structure between the standardized residuals. The Gaussian, Student t, Clayton and Gumbel copula are fitted and so-called ‘blanket’ goodness-of-fit tests are conducted in order to examine the appropriateness of the different copula functions. We also compare the suggested copula models to a standard variance-covariance approach and the Dynamic Conditional Correlation (DCC) model. We find significant positive correlation between the series what somehow contradicts earlier studies on the topic finding negative or only weak positive correlations between REITS and equity investments. Therefore, for Australian markets the diversification potential of investments in REITS might be less significant than for overseas markets. We also find significant tail dependence, in particular in the tail dependence structure that is best modelled by the Student t copula. Conducting a back-testing Value-at-Risk study for a portfolio combining investments in real estate and equity we find that the conditional copula as well as the DCC model significantly outperforms a static variance covariance approach. Our findings suggest that ignoring the complex and dynamic dependence structure in favour of a simple multivariate normal model leads to a significant underestimation of the actual risk.

Modelling the Dependence Structure between Australian Equity and Real Estate Markets – a Copula Approach

In the last decade, the Australian market for Real Estate Investment Trusts (REITS) has shown substantial growth rates. Australian Real Estate Investment Trusts (AREITS) are a unitized portfolio of property assets which allows investors to purchase a share in a diversified and professionally managed portfolio of real estate. We apply conditional copula models in order to investigate the dependence structure between returns of AREITS and Australian equity markets. The dynamic analysis is based on an AR-GARCH approach to model time-varying volatility of the individual series in combination with different copulas to capture the dependence structure between the standardized residuals. The Gaussian, Student t, Clayton and Gumbel copula are fitted and so-called ‘blanket’ goodness-of-fit tests are conducted in order to examine the appropriateness of the different copula functions. We also compare the suggested copula models to a standard variance-covariance approach and a multivariate GARCH-BEKK model. We find significant positive correlation between the series what somehow contradicts earlier studies on the topic finding negative or only weak positive correlations between REITS and equity investments. Therefore, for Australian markets the diversification potential of investments in REITS might be less significant than for overseas markets. We also find significant tail dependence, in particular in the lower left tail that is best modeled by the Clayton copula. Conducting a back testing Value-at-Risk study for a portfolio combining investments in real estate and equity we find that the conditional copula as well as the GARCH BEKK model significantly outperform a static variance covariance approach. Our findings suggest that ignoring the complex and dynamic dependence structure in favor of a simple multivariate normal model leads to a significant underestimation of the actual risk.

Transformation-based nonparametric estimation of multivariate densities

We present a probability-integral-transformation-based estimator of multivariate densities. Given a sample of random vectors, we first transform the data into their corresponding marginal distributions. The marginal densities and the joint density of the transformed data are estimated nonparametrically. The joint density of the original data is constructed as the product of the density of the transformed data and marginal densities, which coincides with the copula representation of multivariate densities. We show that the Kullback–Leibler Information Criterion (KLIC) between the true density and its estimate can be decomposed into the KLIC of the marginal densities and that of the copula density. We derive the convergence rate of the proposed estimator in terms of the KLIC and propose a supervised hierarchical procedure of model selection. Monte Carlo simulations demonstrate the good performance of the estimator. An empirical example on the US and UK stock markets is presented. The estimated conditional copula density provides useful insight into the joint movements of the US and UK markets under extreme Asian markets. • We propose a transformation based nonparametric multivariate density estimator. • We establish convergence in terms of the Kullback–Leibler Information Criterion. • We propose a supervised hierarchical procedure of model specification. • Our simulations demonstrate good performance of the proposed estimator. • Our analysis reveals interesting features of global financial markets.

Exploring copulas for the imputation of complex dependent data

In this work we introduce a copula-based method for imputing missing data by using conditional density functions of the missing variables given the observed ones. In theory, such functions can be derived from the multivariate distribution of the variables of interest. In practice, it is very difficult to model joint distributions and derive conditional distributions, especially when the margins are different. We propose a natural solution to the problem by exploiting copulas so that we derive conditional density functions through the corresponding conditional copulas. The approach is appealing since copula functions enable us (1) to fit any combination of marginal distribution functions, (2) to take into account complex multivariate dependence relationships and (3) to model the marginal distributions and the dependence structure separately. We describe the method and perform a Monte Carlo study in order to compare it with two well-known imputation techniques: the nearest neighbour donor imputation and the regression imputation by EM algorithm. Our results indicate that the proposal compares favourably with classical methods in terms of preservation of microdata, margins and dependence structure.

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.

A copula approach to assessing Granger causality

In neuroscience, as in many other fields of science and engineering, it is crucial to assess the causal interactions among multivariate time series. Granger causality has been increasingly used to identify causal influence between time series based on multivariate autoregressive models. Such an approach is based on linear regression framework with implicit Gaussian assumption of model noise residuals having constant variance. As a consequence, this measure cannot detect the cause-effect relationship in high-order moments and nonlinear causality. Here, we propose an effective model-free, copula-based Granger causality measure that can be used to reveal nonlinear and high-order moment causality. We first formulate Granger causality as the log-likelihood ratio in terms of conditional distribution, and then derive an efficient estimation procedure using conditional copula. We use resampling techniques to build a baseline null-hypothesis distribution from which statistical significance can be derived. We perform a series of simulations to investigate the performance of our copula-based Granger causality, and compare its performance against other state-of-the-art techniques. Our method is finally applied to neural field potential time series recorded from visual cortex of a monkey while performing a visual illusion task.

Additive models for conditional copulas

Conditional copulas are flexible statistical tools that couple joint conditional and marginal conditional distributions. In a linear regression setting with more than one covariate and two dependent outcomes, we consider additive models for studying the dependence between covariates and the copula parameter. We examine the computation and model selection tools needed for Bayesian inference. The method is illustrated using simulations and a real example. Copyright © 2014 John Wiley & Sons, Ltd.