Joint Dependence Structure Research Articles

Robust optimization of mixed CVaR STARR ratio using copulas

We introduce the robust optimization models for two variants of stable tail-adjusted return ratio (STARR), one with mixed conditional value-at-risk (MCVaR) and the other with deviation MCVaR (DMCVaR), under joint ambiguity in the distribution modeled using copulas. The two models are shown to be computationally tractable linear programs. We apply a two-step procedure to capture the joint dependence structure among the assets. We first extract the filtered residuals from the return series of each asset using AutoRegressive Moving Average Glosten Jagannathan Runkle Generalized Autoregressive Conditional Heteroscedastic (ARMA-GJR-GARCH) model. Subsequently, we exploit the regular vine copulas to model the joint dependence among the transformed residuals. The tree structure in the regular vines is accomplished using Kendall’s tau. We compare the performance of the proposed two robust models with their conventional counterparts when the joint distribution in the latter is described using Gaussian copula only. We also examine the performance of the obtained portfolios against those from the Markowitz model and multivariate GARCH models using the rolling window analysis. We illustrate the superior performance of the proposed robust models than their conventional counterpart models on excess mean returns, Sortino ratio, Rachev ratio, VaR ratio, and Treynor ratio, on three data sets comprising of indices across the globe.

A comparison of search strategies to design the cokriging neighborhood for predicting coregionalized variables

Cokriging allows predicting coregionalized variables from sampling information, by considering their spatial joint dependence structure. When secondary covariates are available exhaustively, solving the cokriging equations may become prohibitive, which motivates the use of a moving search neighborhood to select a subset of data, based on their closeness to the target location and the screen effect approximation. This paper investigates the efficiency of different strategies for designing a sub-optimal neighborhood wherein the simplification of the cokriging equations is challenging. To do so, five alternatives (single search, multiple search, strictly collocated search, multi-collocated search and isotopic search) are tested and compared with the reference unique neighborhood, through synthetic examples with different data configurations and spatial joint correlation models. The results indicate that the multi-collocated and multiple searches bear the highest resemblance to the reference case under the analyzed spatial structure models, while the single and the isotopic searches, which do not differentiate the primary and secondary sampling designs, yield the poorest results in terms of cokriging error variance.

Joint modelling of drought characteristics derived from historical and synthetic rainfalls: Application of Generalized Linear Models and Copulas

Study regionÇoruh Basin in Northeastern Turkey. Study focusIn recent years, copulas have been widely used to model the joint distribution function of duration and severity series which are the major characteristics of a drought event to be considered in the planning and management of water resources systems. However, as the copula functions are typically fitted to the drought series that are derived from a limited amount of observed data, it may be insufficient to characterize the full range of the analyzed drought characteristics. Therefore, General Linear Models (GLMs) were used to model and simulate rainfall data in this study. The Standard Precipitation Index (SPI) method was used to obtain the drought characteristics from simulated and historical rainfall series. Four Archimedean copulas, namely Ali-Mikhail-Haq, Clayton, Frank and Gumbel-Hougaard, were evaluated to model the joint distribution functions of these characteristics. New hydrological insights for the regionThe Gumbel-Hougaard copula was found to be the most suitable copula in modelling the joint dependence structure of the drought characteristics at five stations in the basin. The derived Gumbel-Hougaard copulas for each station were employed to obtain joint and conditional return periods of the historical and generated drought characteristics. The drought risks that are estimated based on bivariate return periods for different circumstances can provide useful information in planning, management and in assessing adequacy of the water structures in the basin.

Bivariate frequency analysis of flood and extreme precipitation under changing environment: case study in catchments of the Loess Plateau, China

Floods have changed in a complex manner, triggered by the changing environment (i.e., intensified human activities and global warming). Hence, for better flood control and mitigation in the future, bivariate frequency analysis of flood and extreme precipitation events is of great necessity to be performed within the context of changing environment. Given this, in this paper, the Pettitt test and wavelet coherence transform analysis are used in combination to identify the period with transformed flood-generating mechanism. Subsequently, the primary and secondary return periods of annual maximum flood (AMF) discharge and extreme precipitation (Pr) during the identified period are derived based on the copula. Meanwhile, the conditional probability of occurring different flood discharge magnitudes under various extreme precipitation scenarios are estimated using the joint dependence structure between AMF and Pr. Moreover, Monte Carlo-based algorithm is performed to evaluate the uncertainties of the above copula-based analyses robustly. Two catchments located on the Loess plateau are selected as study regions, which are Weihe River Basin (WRB) and Jinghe River Basin (JRB). Results indicate that: (1) the 1994–2014 and 1981–2014 are identified as periods with transformed flood-generating mechanism in the WRB and JRB, respectively; (2) the primary and secondary return periods for AMF and Pr are examined. Furthermore, chance of occurring different AMF under varying Pr scenarios also be elucidated according to the joint distribution of AMF and Pr. Despite these, one thing to notice is that the associate uncertainties are considerable, thus greatly challenges measures of future flood mitigation. Results of this study offer technical reference for copula-based frequency analysis under changing environment at regional and global scales.

A probabilistic prediction network for hydrological drought identification and environmental flow assessment

AbstractA general probabilistic prediction network is proposed for hydrological drought examination and environmental flow assessment. This network consists of three major components. First, we present the joint streamflow drought indicator (JSDI) to describe the hydrological dryness/wetness conditions. The JSDI is established based on a high‐dimensional multivariate probabilistic model. In the second part, a drought‐based environmental flow assessment method is introduced, which provides dynamic risk‐based information about how much flow (the environmental flow target) is required for drought recovery and its likelihood under different hydrological drought initial situations. The final part involves estimating the conditional probability of achieving the required environmental flow under different precipitation scenarios according to the joint dependence structure between streamflow and precipitation. Three watersheds from different countries (Germany, China, and the United States) with varying sizes from small to large were used to examine the usefulness of this network. The results show that the JSDI can provide an assessment of overall hydrological dryness/wetness conditions and performs well in identifying both drought onset and persistence. This network also allows quantitative prediction of targeted environmental flow required for hydrological drought recovery and estimation of the corresponding likelihood. Moreover, the results confirm that the general network can estimate the conditional probability associated with the required flow under different precipitation scenarios. The presented methodology offers a promising tool for water supply planning and management and for drought‐based environmental flow assessment. The network has no restrictions that would prevent it from being applied to other basins worldwide.

Joint probability of precipitation and reservoir storage for drought estimation in the headwater basin of the Huaihe River, China

Reservoir storage plays an important role in water supply during the dry season when precipitation is insufficient. In a watershed where the streams are controlled by reservoirs, drought occurrences depend on not only precipitation variations but also reservoir regulation. In this study, the joint dependence structure of the reservoir storage and its relevant variables of precipitation and/or upstream outflow were analyzed for two cascade reservoirs in a headwater basin of the Huaihe River, China. Correlation analysis indicates that the reservoir storage in October (the end of the wet season) depends highly on the regional precipitation at time scales of several months, e.g., 7 months for the upstream and 9 months for the downstream. Additionally, the downstream storage is correlated with outflow from the upstream reservoir at the 5-month timescale significantly. For estimation of the joint probability of pairs of the storage and its relevant variables, univariate marginal distributions and bivariate copula were appropriately selected in terms of statistical tests. The bivariate return period of \(T(X < x,Y < y)\) and \(T(X \le x,Y \ge y)\) and the conditional probability of \(P(Y \ge y|X \le x)\) were estimated by using the selected Clayton copula. The results from contour lines of the bivariate return period demonstrate that the probability of drought occurrences affected by both reservoir storage and precipitation/outflow is smaller than that by either of the variables. Meanwhile, the concurrent drought probability between precipitation and reservoir storage in the upstream is higher than that in the downstream. The estimated conditional probability offers useful information on how much the regular storage could be remained under some specified drought levels of precipitation/upstream outflow. Therefore, the results are helpful for improving the operation strategies of the cascade reservoirs for the adaptive management of drought under different climate variations.

Bayesian Graphical Models for Differential Pathways

Graphical models can be used to characterize the dependence structure for a set of random variables. In some applications, the form of dependence varies across different subgroups. This situation arises, for example, when protein activation on a certain pathway is recorded, and a subgroup of patients is characterized by a pathological disruption of that pathway. A similar situation arises when one subgroup of patients is treated with a drug that targets that same pathway. In both cases, understanding changes in the joint distribution and dependence structure across the two subgroups is key to the desired inference. Fitting a single model for the entire data could mask the differences. Separate independent analyses, on the other hand, could reduce the effective sample size and ignore the common features. In this paper, we develop a Bayesian graphical model that addresses heterogeneity and implements borrowing of strength across the two subgroups by simultaneously centering the prior towards a global network. The key feature is a hierarchical prior for graphs that borrows strength across edges, resulting in a comparison of pathways across subpopulations (differential pathways) under a unified model-based framework. We apply the proposed model to data sets from two very different studies: histone modifications from ChIP-seq experiments, and protein measurements based on tissue microarrays.

Linking the gas and oil markets with the stock market: Investigating the U.S. relationship

Energy markets can represent a strategic advantage when they are supporting each other, and specifically when energy segments are complementary enough to support economic development and growth. In this light, a high and strategic interest relies on the possible interactions between energy market segments as well as their impact on a given country’s financial market. The proposed research focuses on the interaction between the U.S. natural gas and U.S. crude oil markets on one side and their dependencies with the U.S. stock market on the other side. After controlling for structural changes or breaks, we characterize previous dependencies with the multivariate copula methodology. First, we assess the joint link prevailing between the natural gas and crude oil markets. Then, we characterize the joint risk structure prevailing between previous energy markets and the U.S. stock market. Finally, we assess the joint dependence structure between the natural gas, crude oil and stock markets.

Comorbidity of chronic diseases in the elderly: Patterns identified by a copula design for mixed responses

Joint modeling of multiple health related random variables is essential to develop an understanding for the public health consequences of an aging population. This is particularly true for patients suffering from multiple chronic diseases. The contribution is to introduce a novel model for multivariate data where some response variables are discrete and some are continuous. It is based on pair copula constructions (PCCs) and has two major advantages over existing methodology. First, expressing the joint dependence structure in terms of bivariate copulas leads to a computationally advantageous expression for the likelihood function. This makes maximum likelihood estimation feasible for large multidimensional data sets. Second, different and possibly asymmetric bivariate (conditional) marginal distributions are allowed which is necessary to accurately describe the limiting behavior of conditional distributions for mixed discrete and continuous responses. The advantages and the favorable predictive performance of the model are demonstrated using data from the Second Longitudinal Study of Aging (LSOA II).

Estimation of Extreme Quantiles for Functions of Dependent Random Variables

Summary We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast with the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The method proposed is semiparametric as no parametric forms are assumed on the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking advantage of the recent development in constructing large dimensional vine copulas. Consequently a sample quantile resulting from a large bootstrap sample drawn from the fitted joint distribution is taken as the estimator for the extreme quantile. This estimator is proved to be consistent under the regularity conditions on the closeness between a quantile set and its truncated set, and the empirical approximation for the truncated set. The simulation results lend further support to the reliable and robust performance of the method proposed. The method is further illustrated by a real world example in backtesting financial risk models.

Correcting for systematic biases in multiple raw GCM variables across a range of timescales

Summary Many hydro-climatological applications require use of General Circulation Models (GCMs) outputs. However, the raw information as available from GCMs often contains significant systematic biases when compared with observations. This necessitates some kind of statistical adjustment to be carried out on the GCM fields before their use in any application. It is common to correct the GCM simulations by removing the systematic biases at the time scale of interest, usually individually for each GCM variable that is needed. The outcome is a set of bias corrected variables that are not assessed for bias in their joint dependence structure, nor biases in all attributes at other time scales (such as daily and monthly and annual) except the one under consideration. In this paper, we present a bias correction approach that simultaneously adjusts for the biases in multiple variables across multiple time scales. The proposed Multivariate Recursive Nesting Bias Correction (MRNBC) approach simultaneously corrects many GCM variables and repeats the procedure across different levels of temporal aggregation to impart observed distributional and persistence properties at multiple time scales. The bias corrected series exhibits improvements across all variables and over all the time scales considered. The use of the approach in hydrology and water resources related downscaling applications is expected to deliver better results.

Copula-Type Estimators for Flexible Multivariate Density Modeling Using Mixtures

Copulas are popular as models for multivariate dependence because they allow the marginal densities and the joint dependence to be modeled separately. However, they usually require that the transformation from uniform marginals to the marginals of the joint dependence structure is known. This can only be done for a restricted set of copulas, for example, a normal copula. Our article introduces copula-type estimators for flexible multivariate density estimation which also allow the marginal densities to be modeled separately from the joint dependence, as in copula modeling, but overcomes the lack of flexibility of most popular copula estimators. An iterative scheme is proposed for estimating copula-type estimators and its usefulness is demonstrated through simulation and real examples. The joint dependence is modeled by mixture of normals and mixture of normal factor analyzer models, and mixture of t and mixture of t-factor analyzer models. We develop efficient variational Bayes algorithms for fitting these in which model selection is performed automatically. Based on these mixture models, we construct four classes of copula-type densities which are far more flexible than current popular copula densities, and outperform them in a simulated dataset and several real datasets. Supplementary material for this article is available online.

Copula based factorization in Bayesian multivariate infinite mixture models

Bayesian nonparametric models based on infinite mixtures of density kernels have been recently gaining in popularity due to their flexibility and feasibility of implementation even in complicated modeling scenarios. However, these models have been rarely applied in more than one dimension. Indeed, implementation in the multivariate case is inherently difficult due to the rapidly increasing number of parameters needed to characterize the joint dependence structure accurately. In this paper, we propose a factorization scheme of multivariate dependence structures based on the copula modeling framework, whereby each marginal dimension in the mixing parameter space is modeled separately and the marginals are then linked by a nonparametric random copula function. Specifically, we consider nonparametric univariate Gaussian mixtures for the marginals and a multivariate random Bernstein polynomial copula for the link function, under the Dirichlet process prior. We show that in a multivariate setting this scheme leads to an improvement in the precision of a density estimate relative to the commonly used multivariate Gaussian mixture. We derive weak posterior consistency of the copula-based mixing scheme for general kernel types under high-level conditions, and strong posterior consistency for the specific Bernstein–Gaussian mixture model.

Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas

In this paper, a copula based methodology is presented for flood frequency analysis of Upper Godavari River flows in India. By using the specific advantages of copula method in modeling the joint dependence structure of uncertain variables, this study applies Archimedean copulas for frequency analysis of flood characteristics annual peak flow, flood volume and flood duration. To determine the best fit marginal distributions for flood variables, few parametric and nonparametric probability distributions are examined and the best fit model is adopted for copula modeling. Four Archimedean family of copulas, namely Ali-Mikhail-Haq, Clayton, Gumbel-Hougaard and Frank copulas are evaluated for modeling the joint dependence of annual peak flow-volume, and flood volume-duration pairs. The performance of two parameter estimation methods, namely method-of-moments-like estimator based on inversion of Kendall’s tau and maximum pseudo-likelihood estimator for copulas are investigated. On performing Monte Carlo simulation to assess the performance of copula distributions in modeling the joint dependence structure of flood variables, it is found that the developed copula models are well representing the observed flood characteristics. From standard statistical tests, Frank copula has been identified as the best fitted copula for both bivariate models. The Frank copula function is used for obtaining joint and conditional return periods of flood characteristics, which can be useful for risk based design of water resources projects.

Risk Assessment of Droughts in Gujarat Using Bivariate Copulas

This study presents risk assessment of hydrologic extreme events droughts in Saurashtra and Kutch region of Gujarat state, India. Drought is a recurrent phenomenon and risk assessment of droughts can play an important role in proper planning and management of water resources in the study region. In the study, drought events are characterized by severity and duration, and drought occurrences are modeled by Standardized Precipitation Index (SPI) computed on mean areal precipitation, aggregated at a time scale of 6 months for the period 1900–2008. After evaluating several distribution functions, drought variable—severity is best described by non-parametric kernel density, whereas duration is best fitted by exponential distribution. Considering the extreme nature of drought variables, the upper tail dependence copula families including two Archimedean—Gumbel-Hougaard, BB1 and one elliptical—Student’s t copulas are evaluated for modeling joint distribution of drought variables. On evaluating their performance using various goodness-of-fit measures, Gumbel-Hougaard copula is found to be the best performing copula in modeling the joint dependence structure of drought variables. Also, while comparing with traditional bivariate distributions, the copula based distributions are resulted in better performance as compared to bivariate log-normal and the logistic model for bivariate extreme value distributions. Then joint and conditional return periods of drought characteristics are derived, which can be helpful for risk based planning and management of water resources systems in the study region.

Exchange Rates Dependence: What Drives It?

Exchange rate movements are difficult to predict but there appear to be discernible patterns in how currencies jointly appreciate or depreciate against the dollar. In this paper, we study the dependence structure of a number of exchange rate pairs against the dollar. We employ a conditional copula approach to recover the joint distributions for pairs of exchange rates and study both the correlation and the upper and lower tail dependence of these distributions. We analyze changes in dependence measures over time, and we investigate whether these measures are affected by the business cycle or interest rate differentials. Our results show that dependencies are indeed time-varying. We find that foreign and U.S. recessions affect the joint dependence structure and that currencies with higher interest rate differentials tend to move less closely together, not only on average (correlation), but also when extreme events occur (tails).

Exchange Rates Dependence: What Drives It?

Exchange rate movements are difficult to predict but there appear to be discernible patterns in how currencies jointly appreciate or depreciate against the dollar. In this paper, we study the dependence structure of a number of exchange rate pairs against the dollar. We employ a conditional copula approach to recover the joint distributions for pairs of exchange rates and study both the correlation and the upper and lower tail dependence of these distributions. We analyze changes in dependence measures over time, and we investigate whether these measures are affected by the business cycle or interest rate differentials. Our results show that dependencies are indeed time-varying. We find that foreign and U.S. recessions affect the joint dependence structure and that currencies with higher interest rate differentials tend to move less closely together, not only on average (correlation), but also when extreme events occur (tails).

Stepwise Multiple Testing as Formalized Data Snooping

In econometric applications, often several hypothesis tests are carried out at once. The problem then becomes how to decide which hypotheses to reject, accounting for the multitude of tests. This paper suggests a stepwise multiple testing procedure that asymptotically controls the familywise error rate. Compared to related single-step methods, the procedure is more powerful and often will reject more false hypotheses. In addition, we advocate the use of studentization when feasible. Unlike some stepwise methods, the method implicitly captures the joint dependence structure of the test statistics, which results in increased ability to detect false hypotheses. The methodology is presented in the context of comparing several strategies to a common benchmark. However, our ideas can easily be extended to other contexts where multiple tests occur. Some simulation studies show the improvements of our methods over previous proposals. We also provide an application to a set of real data.

Stepwise Multiple Testing as Formalized Data Snooping

It is common in econometric applications that several hypothesis tests are carried out at the same time. The problem then becomes how to decide which hypotheses to reject, accounting for the multitude of tests. In this paper, we suggest a stepwise multiple testing procedure which asymptotically controls the familywise error rate at a desired level. Compared to related single-step methods, our procedure is more powerful in the sense that it often will reject more false hypotheses. Unlike some stepwise methods, our method implicitly captures the joint dependence structure of the test statistics, which results in increased ability to detect alternative hypotheses. We prove our method asymptotically controls the familywise error rate under minimal assumptions. Some simulation studies show the improvements of our methods over previous proposals. We also provide an application to a set of real data.