Vine copula models with GLM and sparsity

Vine 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.     

Counterdiagonal/nonpositive tail dependence in Vine copula constructions: application to portfolio management

Statistical Methods & Applications - Accurately modelling the dependence structure between financial assets in a portfolio optimization framework has attracted growing attention in statistical...     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

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.     

Directional dependence via Gaussian copula beta regression model with asymmetric GARCH marginals

This article proposes a new directional dependence by using the Gaussian copula beta regression model. In particular, we consider an asymmetric Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) model for the marginal distribution of standardized residuals to make data exhibiting conditionally heteroscedasticity to white noise process. With the simulated data generated by an asymmetric bivariate copula, we verify our proposed directional dependence method. For the multivariate direction dependence by using the Gaussian copula beta regression model, we employ a three-dimensional archemedian copula to generate trivariate data and then show the directional dependence for one random variable given two other random variables. With West Texas Intermediate Daily Price (WTI) and the Standard & Poor’s 500 (S&P 500), our proposed directional dependence by the Gaussian copula beta regression model reveals that the directional dependence from WTI to S&P 500 is greater than that from S&P 500 to WTI. To validate our empirical result, the Granger causality test is conducted, confirming the same result produced by our method.     

Copula representation of bivariate L-moments: a new estimation method for multiparameter two-dimensional copula models

Serfling and Xiao [A contribution to multivariate L-moments, L-comoment matrices. J Multivariate Anal. 2007;98:1765–1781] extended the L-moment theory to the multivariate setting. In the present paper, we focus on the two-dimensional random vectors to establish a link between the bivariate L-moments (BLM) and the underlying bivariate copula functions. This connection provides a new estimate of dependence parameters of bivariate statistical data. Extensive simulation study is carried out to compare estimators based on the BLM, the maximum likelihood, the minimum distance and a rank approximate Z-estimation. The obtained results show that, when the sample size increases, BLM-based estimation performs better as far as the bias and computation time are concerned. Moreover, the root-mean-squared error is quite reasonable and less sensitive in general to outliers than those of the above cited methods. Further, the proposed BLM method is an easy-to-use tool for the estimation of multiparameter copula models. A generalization of the BLM estimation method to the multivariate case is discussed.     

Stochastic approximation Monte Carlo importance sampling for approximating exact conditional probabilities

Importance sampling and Markov chain Monte Carlo methods have been used in exact inference for contingency tables for a long time, however, their performances are not always very satisfactory. In this paper, we propose a stochastic approximation Monte Carlo importance sampling (SAMCIS) method for tackling this problem. SAMCIS is a combination of adaptive Markov chain Monte Carlo and importance sampling, which employs the stochastic approximation Monte Carlo algorithm (Liang et al., J. Am. Stat. Assoc., 102(477):305–320, 2007) to draw samples from an enlarged reference set with a known Markov basis. Compared to the existing importance sampling and Markov chain Monte Carlo methods, SAMCIS has a few advantages, such as fast convergence, ergodicity, and the ability to achieve a desired proportion of valid tables. The numerical results indicate that SAMCIS can outperform the existing importance sampling and Markov chain Monte Carlo methods: It can produce much more accurate estimates in much shorter CPU time than the existing methods, especially for the tables with high degrees of freedom.     

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACT

We establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.     

Estimation of conditional quantiles by a new smoothing approximation of asymmetric loss functions

In this paper, nonparametric estimation of conditional quantiles of a nonlinear time series model is formulated as a nonsmooth optimization problem involving an asymmetric loss function. This asymmetric loss function is nonsmooth and is of the same structure as the so-called lopsided absolute value function. Using an effective smoothing approximation method introduced for this lopsided absolute value function, we obtain a sequence of approximate smooth optimization problems. Some important convergence properties of the approximation are established. Each of these smooth approximate optimization problems is solved by an optimization algorithm based on a sequential quadratic programming approximation with active set strategy. Within the framework of locally linear conditional quantiles, the proposed approach is compared with three other approaches, namely, an approach proposed by Yao and Tong (1996), the Iteratively Reweighted Least Squares method and the Interior-Point method, through some empirical numerical studies using simulated data and the classic lynx pelt series. In particular, the empirical performance of the proposed approach is almost identical with that of the Interior-Point method, both methods being slightly better than the Iteratively Reweighted Least Squares method. The Yao-Tong approach is comparable with the other methods in the ideal cases for the Yao-Tong method, but otherwise it is outperformed by other approaches. An important merit of the proposed approach is that it is conceptually simple and can be readily applied to parametrically nonlinear conditional quantile estimation.     

A dynamic double asymmetric copula generalized autoregressive conditional heteroskedasticity model: application to China's and US stock market

Modeling the relationship between multiple financial markets has had a great deal of attention in both literature and real-life applications. One state-of-the-art technique is that the individual financial market is modeled by generalized autoregressive conditional heteroskedasticity (GARCH) process, while market dependence is modeled by copula, e.g. dynamic asymmetric copula-GARCH. As an extension, we propose a dynamic double asymmetric copula (DDAC)-GARCH model to allow for the joint asymmetry caused by the negative shocks as well as by the copula model. Furthermore, our model adopts a more intuitive way of constructing the sample correlation matrix. Our new model yet satisfies the positive-definite condition as found in dynamic conditional correlation-GARCH and constant conditional correlation-GARCH models. The simulation study shows the performance of the maximum likelihood estimate for DDAC-GARCH model. As a case study, we apply this model to examine the dependence between China and US stock markets since 1990s. We conduct a series of likelihood ratio test tests that demonstrate our extension (dynamic double joint asymmetry) is adequate in dynamic dependence modeling. Also, we propose a simulation method involving the DDAC-GARCH model to estimate value at risk (VaR) of a portfolio. Our study shows that the proposed method depicts VaR much better than well-established variance–covariance method.     

On the diversity score: a copula approach

ABSTRACT

For the rating process of Collateralized Debt Obligations', Moody's suggests the Diversity Score as a measure of diversification in the collateral pool. This measure is used in Moody's Binomial Expansion Technique to infer the probability of default and thus the expected Loss in the portfolio. In this paper, we examine the appropriateness of this approach to assess the reality of defaults using a copula approach and lower tail dependence.     

A Bayesian method for query approximation

This study presents statistical techniques to obtain local approximate query answers for aggregate multivariate materialized views thus eliminating the need for repetitive scanning of the source data. In widely distributed management information systems, detailed data do not necessarily reside in the same physical location as the decision-maker; thus, requiring scanning of the source data as needed by the query demand. Decision-making, business intelligence and data analysis could involve multiple data sources, data diversity, aggregates and large amounts of data. Management often confronts delays in information acquisition from remote sites. Management decisions usually involve analyses that require the most precise summary data available. These summaries are readily available from data warehouses and can be used to estimate or approximate data in exchange for a quicker response. An approach to supporting aggregate materialized view management is proposed that reconstructs data sets locally using posterior parameter estimates based on sufficient statistics in a log-linear model with a multinomial likelihood.     

Regression in a copula model for bivariate count data

In many cases of modeling bivariate count data, the interest lies on studying the association rather than the marginal properties. We form a flexible regression copula-based model where covariates are used not only for the marginal but also for the copula parameters. Since copula measures the association, the use of covariates in its parameters allow for direct modeling of association. A real-data application related to transaction market basket data is used. Our goal is to refine and understand whether the association between the number of purchases of certain product categories depends on particular demographic customers’ characteristics. Such information is important for decision making for marketing purposes.     

On a new positive dependence concept based on the conditional mean inactivity time order

In this paper, we introduce a new positive dependence concept between two non negative random variables which is related to a conditional version of the mean inactivity time order. A number of properties and relationship between the new notion and the concept of positive likelihood ratio dependence (PLRD) is discussed. Some results in terms of proposed notions for the Archimedean family of copulas are provided.     

Correction to: Joint and conditional dependence modelling of peak district heating demand and outdoor temperature: a copula-based approach

Statistical Methods & Applications - This paper examines the complex dependence between peak district heating demand and outdoor temperature. Our aim is to provide the probability law of heat...     

Goodness-of-fit for regime-switching copula models with application to option pricing

We consider several time series, and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modelled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit procedure based on Cramér–von Mises statistics is proposed to select the appropriate number of regimes. Numerical experiments are performed to assess the validity of the proposed methodology. As an example of application, we evaluate a European put-on-max option on the returns of two assets. To facilitate the use of our methodology, we have built a R package HMMcopula available on CRAN. The Canadian Journal of Statistics 48: 79–96; 2020 © 2020 Statistical Society of Canada     

The stochastic approximation method for the estimation of a multivariate probability density

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a probability density, including the one introduced by Hall and Patil [1994. On the efficiency of on-line density estimators. IEEE Trans. Inform. Theory 40, 1504–1512]. We study the properties of these estimators and compare them with Rosenblatt's nonrecursive estimator. It turns out that, for pointwise estimation, it is preferable to use the nonrecursive Rosenblatt's kernel estimator rather than any recursive estimator. A contrario, for estimation by confidence intervals, it is better to use a recursive estimator rather than Rosenblatt's estimator.