Underlying Dependence Structure Research Articles

Goodness-of-fit tests for the family of multivariate chi-square copulas

Nonparametric moment-based goodness-of-fit tests are developed for the family of chi-square copulas of arbitrary dimensions. This class of dependence models allows for tail asymmetries and contains the family of multivariate normal copulas as a special case. The proposed tests are based on two rank correlation coefficients whose population versions are equal, up to a monotone transformation, when the underlying dependence structure is a chi-square copula. The test statistics are computed from natural rank-based estimations of these two correlation coefficients and their large-sample distributions under the null hypothesis of a chi-square copula are derived; the validity of a parametric bootstrap procedure for the computation of p-values is formally established as well. Particular attention is given to tests for the families of normal and centered chi-square copulas. The simulations that are reported indicate that the new tests are reliable alternatives to those based on the empirical copula, both in the bivariate and multivariate cases. The usefulness of the introduced methodology is illustrated on the five-dimensional Nutrient dataset.

Multi-Panel Kendall plot in light of an ROC curve analysis applied to measuring dependence

ABSTRACTThe Kendall plot (-plot) is a plot measuring dependence between the components of a bivariate random variable. The -plot graphs the Kendall distribution function against the distribution function of VU, where V and U are independent uniform random variables. We associate -plots with the receiver operating characteristic () curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the curve (). In parallel with the , we propose a novel strategy to measure association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve () cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the -plots, and define an index of dependence that is based on . This measure characterizes a wide range of two-variable relationships, thereby completely detecting the underlying dependence structure. Properties of the proposed index satisfy the mathematical definition of a measure. Finally, simulations and real data examples illustrate the applicability of the proposed method.

Modeling the Dependence Structure of Share Prices among Three Chinese City Banks

We study the dependence structure of share price returns among the Beijing Bank, Ningbo Bank, and Nanjing Bank using copula models. We use the normal, Student’s t, rotated Gumbel, and symmetrized Joe-Clayton (SJC) copula models to estimate the underlying dependence structure in two periods: one covering the global financial crisis and the other covering the domestic share market crash in China. We show that Beijing Bank is less dependent on the other two city banks than Nanjing Bank, which is dependent on the other two in share price extreme returns. We also observe a major decrease of dependency from 2007 to 2018 in three one-to-one dependence structures. Interestingly, contrary to recent literatures, Ningbo Bank and Nanjing Bank tend to be more dependent on each other in positive returns than in negative returns during the past decade. We also show the dynamic dependence structures among three city banks using time-varying copula.

Investigating dependence between frequency and severity via simple generalized linear models

Recently, a body of literature proposed new models relaxing a widely-used but controversial assumption of independence between claim frequency and severity in non-life insurance rate making. This paper critically reviews a generalized linear model approach, where a dependence between claim frequency and severity is introduced by treating frequency as a covariate in a regression model for severity. As an extension of this approach, we propose a dispersion model for severity. For this model, the information loss caused by using average severity rather than individual severity is examined in detail and the parameter estimators suffering from low efficiency are identified. We also provide analytical solutions for the aggregate sum to help rate making. We show that the simple functional form used in current research may not properly reflect the real underlying dependence structure. A real data analysis is given to explain our analytical findings.

A semiparametric additive rate model for a modulated renewal process.

Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.

Insurance Portfolio Risk Retention

ABSTRACTIn this article, I introduce a statistic for managing a portfolio of insurance risks. This tool is based on changes in the risk profile when changes in a risk parameter, such as a deductible, coinsurance, or upper policy limit, are made. I refer to the new statistic as a risk measure relative marginal change and denote it as RM2. By examining data from the Wisconsin Local Government Property Fund, I show how it can be used by an insurer to identify the “best” and “worst” risks in terms of opportunities for risk management. The RM2 changes reflect the underlying dependence structure of risks; I use an elliptical copula framework to demonstrate the sensitivity of risk mitigation strategy to the dependence structure.

Path-consistent wrong-way risk: a structural model approach

We present a general and path-consistent wrong-way risk (WWR) model, which does not require simulation of credit and market variables simultaneously. Although similar so-called copula models are well known, our approach is novel in several ways. First, our method can model a wide range of dependence structures while always guaranteeing path consistency of default probabilities (the possibility of path-inconsistencies in copula models was highlighted in a recent article). Second, we place special emphasis on the difficult task of calibrating the underlying dependence structure. In particular, we consider a default correction of the dependence structure. Third, our model serves as a bridge between structural model approaches, where dependence between exposure and equity price is modeled, and copula models, where exposure is directly correlated to default time. Finally, we apply our method in realistic situations and show that we can achieve a wide range of WWR impacts.

On conditional skewness with applications to environmental data

The statistical literature contains many univariate and multivariate skewness measures that allow two datasets to be compared, some of which are defined in terms of quantile values. In most situations, the comparison between two random vectors focuses on univariate comparisons of conditional random variables truncated in quantiles; this kind of comparison is of particular interest in the environmental sciences. In this work, we describe a new approach to comparing skewness in terms of the univariate convex transform ordering proposed by van Zwet (Convex transformations of random variables. Mathematical Centre Tracts, Amsterdam, 1964), associated with skewness as well as concentration. The key to these comparisons is the underlying dependence structure of the random vectors. Below we describe graphical tools and use several examples to illustrate these comparisons.

Modelling structured data with Probabilistic Graphical Models

Most clustering and classification methods are based on the assumption that the objects to be clustered are independent. However, in more and more modern applications, data are structured in a way that makes this assumption not realistic and potentially misleading. A typical example that can be viewed as a clustering task is image segmentation where the objects are the pixels on a regular grid and depend on neighbouring pixels on this grid. Also, when data are geographically located, it is of interest to cluster data with an underlying dependence structure accounting for some spatial localisation. These spatial interactions can be naturally encoded via a graph not necessarily regular as a grid. Data sets can then be modelled via Markov random fields and mixture models (e.g. the so-called MRF and Hidden MRF). More generally, probabilistic graphical models are tools that can be used to represent and manipulate data in a structured way while modeling uncertainty. This chapter introduces the basic concepts. The two main classes of probabilistic graphical models are considered: Bayesian networks and Markov networks. The key concept of conditional independence and its link to Markov properties is presented. The main problems that can be solved with such tools are described. Some illustrations are given associated with some practical work.

A Graphical Goodness-of-Fit Test for Dependence Models in Higher Dimensions

This article introduces a graphical goodness-of-fit test for copulas in more than two dimensions. The test is based on pairs of variables and can thus be interpreted as a first-order approximation of the underlying dependence structure. The idea is to first transform pairs of data columns with the Rosenblatt transform to bivariate standard uniform distributions under the null hypothesis. This hypothesis can be graphically tested with a matrix of bivariate scatterplots, Q-Q plots, or other transformations. Furthermore, additional information can be encoded as background color, such as measures of association or (approximate) p-values of tests of independence. The proposed goodness-of-fit test is designed as a basic graphical tool for detecting deviations from a postulated, possibly high-dimensional, dependence model. Various examples are given and the methodology is applied to a financial dataset. An implementation is provided by the R package copula. Supplementary material for this article is available online, which provides the R package copula and reproduces all the graphical results of this article.

The Improbable Nature of the Implied Correlation Matrix from Spatial Regression Models

Spatial lag dependence in a regression model is similar to the inclusion of a serially autoregressive term for the dependent variable in a time-series context. However, unlike in the time-series model, the implied covariance structure matrix from the spatial autoregressive model can have a very counterintuitive and improbable structure. A single value of spatial autocorrelation parameter can imply a large band of values of pair-wise correlations among different observations of the dependent variable, when the weight matrix for the spatial model is specified exogenously. This is illustrated using cigarette sales data (1963-1992) of 46 US states. It can be seen that that two close neighbours can have very low implied correlations compared to distant neighbours when the weighting scheme is the first-order contiguity matrix. However, if the weight matrix can capture the underlying dependence structure of the observations, then this unintuitive behaviour of implied correlation is corrected to a large extent. From this, the possibility of constructing the weight matrix (or the overall spatial dependence in the data) that is consistent with the underlying correlation structure of the dependent variable is explored. The suggested procedures produced very positive results indicating further research.

Pricing equity and debt tranches of collateralized funds of hedge fund obligations: An approach based on stochastic time change and Esscher-transformed martingale measure

Collateralized Funds of Hedge Fund Obligations (CFOs) are relatively recent structured finance products linked to the performance of underlying funds of hedge funds. The capital structure of a CFO is similar to traditional Collateralized Debt Obligations (CDOs), meaning that investors are offered different rated notes and equity interests. CFOs are structured as arbitrage market value CDOs. The fund of funds manager actively manages the fund to maximize total returns while limiting price volatility within the guidelines of the structure. The aim of this paper is to provide a useful framework to evaluate Collateralized Funds of Hedge Fund Obligations, that is pricing the equity and the debt tranches of a CFO. The basic idea is to evaluate each CFO liability as an option written on the underlying pool of hedge funds. The value of every tranche depends on the evolution of the collateral portfolio during the life of the contract. Care is taken to distinguish between the fund of hedge funds and its underlying hedge funds, each of which is itself a portfolio of various securities, debt instruments and financial contracts. The proposed model incorporates skewness, excess kurtosis and is able to capture more complex dependence structures among hedge fund log-returns than the mere correlation matrix. With this model it is possible to describe the impact of an equivalent change of probability measure not only on the marginal processes but also on the underlying dependence structure among hedge funds.

Conditional copula simulation for systemic risk stress testing

Since the financial crisis of 2007–2009 there is an active debate by regulators and academic researchers on systemic risk, with the aim of preventing similar crises in the future or at least reducing their impact. A major determinant of systemic risk is the interconnectedness of the international financial market. We propose to analyze interdependencies in the financial market using copulas, in particular using flexible vine copulas, which overcome limitations of the popular elliptical and Archimedean copulas. To investigate contagion effects among financial institutions, we develop methods for stress testing by exploiting the underlying dependence structure. New approaches for Archimedean and, especially, for vine copulas are derived. In a case study of 38 major international institutions, 20 insurers and 18 banks, we then analyze interdependencies of CDS spreads and perform a systemic risk stress test. The specified dependence model and the results from the stress test provide new insights into the interconnectedness of banks and insurers. In particular, the failure of a bank seems to constitute a larger systemic risk than the failure of an insurer.

Memory and other properties of multiple test procedures generated by entangled graphs

Methods for addressing multiplicity in clinical trials have attracted much attention during the past 20 years. They include the investigation of new classes of multiple test procedures, such as fixed sequence, fallback and gatekeeping procedures. More recently, sequentially rejective graphical test procedures have been introduced to construct and visualize complex multiple test strategies. These methods propagate the local significance level of a rejected null hypothesis to not-yet rejected hypotheses. In the graph defining the test procedure, hypotheses together with their local significance levels are represented by weighted vertices and the propagation rule by weighted directed edges. An algorithm provides the rules for updating the local significance levels and the transition weights after rejecting an individual hypothesis. These graphical procedures have no memory in the sense that the origin of the propagated significance level is ignored in subsequent iterations. However, in some clinical trial applications, memory is desirable to reflect the underlying dependence structure of the study objectives. In such cases, it would allow the further propagation of significance levels to be dependent on their origin and thus reflect the grouped parent-descendant structures of the hypotheses. We will give examples of such situations and show how to induce memory and other properties by convex combination of several individual graphs. The resulting entangled graphs provide an intuitive way to represent the underlying relative importance relationships between the hypotheses, are as easy to perform as the original individual graphs, remain sequentially rejective and control the familywise error rate in the strong sense.

A Simulation-Based Approach to Decision Making with Partial Information

The construction of a probabilistic model is a key step in most decision and risk analyses. Typically this is done by defining a single joint distribution in terms of marginal and conditional distributions. The difficulty of this approach is that often the joint distribution is underspecified. For example, we may lack knowledge of the marginal distributions or the underlying dependence structure. In this paper, we suggest an approach to analyzing decisions with partial information. Specifically, we propose a simulation procedure to create a collection of joint distributions that match the known information. This collection of distributions can then be used to analyze the decision problem. We demonstrate our method by applying it to the Eagle Airlines case study used in previous studies.

Analysis of high dimensional data using pre-defined set and subset information, with applications to genomic data

BackgroundBased on available biological information, genomic data can often be partitioned into pre-defined sets (e.g. pathways) and subsets within sets. Biologists are often interested in determining whether some pre-defined sets of variables (e.g. genes) are differentially expressed under varying experimental conditions. Several procedures are available in the literature for making such determinations, however, they do not take into account information regarding the subsets within each set. Secondly, variables (e.g. genes) belonging to a set or a subset are potentially correlated, yet such information is often ignored and univariate methods are used. This may result in loss of power and/or inflated false positive rate.ResultsWe introduce a multiple testing-based methodology which makes use of available information regarding biologically relevant subsets within each pre-defined set of variables while exploiting the underlying dependence structure among the variables. Using this methodology, a biologist may not only determine whether a set of variables are differentially expressed between two experimental conditions, but may also test whether specific subsets within a significant set are also significant.ConclusionsThe proposed methodology; (a) is easy to implement, (b) does not require inverting potentially singular covariance matrices, and (c) controls the family wise error rate (FWER) at the desired nominal level, (d) is robust to the underlying distribution and covariance structures. Although for simplicity of exposition, the methodology is described for microarray gene expression data, it is also applicable to any high dimensional data, such as the mRNA seq data, CpG methylation data etc.

Conditional dependence structure between oil prices and exchange rates: A copula-GARCH approach

We study the conditional dependence structure between crude oil prices and U.S. dollar exchange rates using a copula-GARCH approach. Various copula functions of the elliptical, Archimedean and quadratic families are used to model the underlying dependence structure in both bearish and bullish market phases. Over the 2000–2011 period, we find evidence of significant and symmetric dependence for almost all the oil-exchange rate pairs considered. The rise in the price of oil is found to be associated with the depreciation of the dollar. Moreover, we show that Student-t copulas best capture the extreme dependence, and that taking the extreme comovement into account leads to improve the accuracy of VaR forecasts. Our main results remain unchanged when considering alternative GARCH-type specifications and the crisis period, but are sensitive to the use of raw returns.

Spatial dependence estimation using FFT of biased covariances

One of the main problems in geostatistics is fitting a valid variogram or covariogram model in order to describe the underlying dependence structure in the data. The dependence between observations can be also modeled in the spectral domain, but the traditional methods based on the periodogram as an estimator of the spectral density may present some problems for the spatial case. In this work, we propose an estimation method for the covariogram parameters based on the fast Fourier transform (FFT) of biased covariances. The performance of this estimator for finite samples is compared through a simulation study with other classical methods stated in spatial domain, such as weighted least squares and maximum likelihood, as well as with other spectral estimators. Additionally, an example of application to real data is given.

Dependence structure between the equity market and the foreign exchange market–A copula approach

This paper investigates the dependence structure between the equity market and the foreign exchange market by using copulas. In particular, several copulas with different dependence structure are compared and used to directly model the underlying dependence structure. We find that there exists significant symmetric upper and lower tail dependence between the two financial markets, and the dependence remains significant but weaker after the launch of the euro. Our findings have important implications for both global investment risk management and international asset pricing by taking into account joint tail risk.

On the construction of stationary AR(1) models via random distributions

We explore a method for constructing first-order stationary autoregressive-type models with given marginal distributions. We impose the underlying dependence structure in the model using Bayesian non-parametric predictive distributions. This approach allows for nonlinear dependency and at the same time works for any choice of marginal distribution. In particular, we look at the case of discrete-valued models; that is the marginal distributions are supported on the non-negative integers.