Asymmetric Forecast Densities for U.S. Macroeconomic Variables from a Gaussian Copula Model of Cross-Sectional and Serial Dependence

Most existing reduced-form macroeconomic multivariate time series models employ elliptical disturbances, so that the forecast densities produced are symmetric. In this article, we use a copula model with asymmetric margins to produce forecast densities with the scope for severe departures from symmetry. Empirical and skew t distributions are employed for the margins, and a high-dimensional Gaussian copula is used to jointly capture cross-sectional and (multivariate) serial dependence. The copula parameter matrix is given by the correlation matrix of a latent stationary and Markov vector autoregression (VAR). We show that the likelihood can be evaluated efficiently using the unique partial correlations, and estimate the copula using Bayesian methods. We examine the forecasting performance of the model for four U.S. macroeconomic variables between 1975:Q1 and 2011:Q2 using quarterly real-time data. We find that the point and density forecasts from the copula model are competitive with those from a Bayesian VAR. During the recent recession the forecast densities exhibit substantial asymmetry, avoiding some of the pitfalls of the symmetric forecast densities from the Bayesian VAR. We show that the asymmetries in the predictive distributions of GDP growth and inflation are similar to those found in the probabilistic forecasts from the Survey of Professional Forecasters. Last, we find that unlike the linear VAR model, our fitted Gaussian copula models exhibit nonlinear dependencies between some macroeconomic variables. This article has online supplementary material.     

Gaussian copula distributions for mixed data,with application in discrimination

The construction of a joint model for mixed discrete and continuous random variables that accounts for their associations is an important statistical problem in many practical applications. In this paper, we use copulas to construct a class of joint distributions of mixed discrete and continuous random variables. In particular, we employ the Gaussian copula to generate joint distributions for mixed variables. Examples include the robit-normal and probit-normal-exponential distributions, the first for modelling the distribution of mixed binary-continuous data and the second for a mixture of continuous, binary and trichotomous variables. The new class of joint distributions is general enough to include many mixed-data models currently available. We study properties of the distributions and outline likelihood estimation; a small simulation study is used to investigate the finite-sample properties of estimates obtained by full and pairwise likelihood methods. Finally, we present an application to discriminant analysis of multiple correlated binary and continuous data from a study involving advanced breast cancer patients.     

Eliciting Dirichlet and Gaussian copula prior distributions for multinomial models

In this paper, we propose novel methods of quantifying expert opinion about prior distributions for multinomial models. Two different multivariate priors are elicited using median and quartile assessments of the multinomial probabilities. First, we start by eliciting a univariate beta distribution for the probability of each category. Then we elicit the hyperparameters of the Dirichlet distribution, as a tractable conjugate prior, from those of the univariate betas through various forms of reconciliation using least-squares techniques. However, a multivariate copula function will give a more flexible correlation structure between multinomial parameters if it is used as their multivariate prior distribution. So, second, we use beta marginal distributions to construct a Gaussian copula as a multivariate normal distribution function that binds these marginals and expresses the dependence structure between them. The proposed method elicits a positive-definite correlation matrix of this Gaussian copula. The two proposed methods are designed to be used through interactive graphical software written in Java.     

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.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

Matching a correlation coefficient by a Gaussian copula

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler’s formula, a polynomial of ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.     

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

Doubly inflated Poisson model using Gaussian copula

Multivariate data are present in many research areas. Its analysis is challenging when assumptions of normality are violated and the data are discrete. The Poisson discrete data can be thought of as very common discrete type, but the inflated and the doubly inflated correspondence are gaining popularity (Sengupta, Chaganty, and Sabo 2015; Lee, Jung, and Jin 2009; Agarwal, Gelfand, and Citron-Pousty 2002).

Our aim is to build a statistical model that can be tractable and used to estimate the model parameters for the multivariate doubly inflated Poisson. To keep the correlation structure, we incorporate ideas from the copula distributions. A multivariate doubly inflated Poisson distribution using Gaussian copula is introduced. Data simulation and parameter estimation algorithms are also provided. Residual checks are carried out to assess any substantial biases. The model dimensionality has been increased to test the performance of the provided estimation method. All results show high-efficiency and promising outcomes in the modeling of discrete data and particularly the doubly inflated Poisson count type data, under a novel modified algorithm.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Model-based clustering of Gaussian copulas for mixed data

Clustering of mixed data is important yet challenging due to a shortage of conventional distributions for such data. In this article, we propose a mixture model of Gaussian copulas for clustering mixed data. Indeed copulas, and Gaussian copulas in particular, are powerful tools for easily modeling the distribution of multivariate variables. This model clusters data sets with continuous, integer, and ordinal variables (all having a cumulative distribution function) by considering the intra-component dependencies in a similar way to the Gaussian mixture. Indeed, each component of the Gaussian copula mixture produces a correlation coefficient for each pair of variables and its univariate margins follow standard distributions (Gaussian, Poisson, and ordered multinomial) depending on the nature of the variable (continuous, integer, or ordinal). As an interesting by-product, this model generalizes many well-known approaches and provides tools for visualization based on its parameters. The Bayesian inference is achieved with a Metropolis-within-Gibbs sampler. The numerical experiments, on simulated and real data, illustrate the benefits of the proposed model: flexible and meaningful parameterization combined with visualization features.     

Posterior moments of scalar functions of a covariance matrix

Given multivariate normal data and a certain spherically invariant prior distribution on the covariance matrix, it is desired to estimate the moments of the posterior marginal distributions of some scalar functions of the covariance matrix by importance sampling. To this end a family of distributions is defined on the group of orthogonal matrices and a procedure is proposed for selecting one of these distributions for use as a weighting distribution in the importance sampling process. In an example estimates are calculated for the posterior mean and variance of each element in the covariance matrix expressed in the original coordinates, for the posterior mean of each element in the correlation matrix expressed in the original coordinates, and for the posterior mean of each element in the covariance matrix expressed in the coordinates of the principal variables.     

A Family of Goodness‐of‐Fit Tests for Copulas Based on Characteristic Functions

A general class of rank statistics based on the characteristic function is introduced for testing goodness‐of‐fit hypotheses about the copula of a continuous random vector. These statistics are defined as L ₂ weighted functional distances between a nonparametric estimator and a semi‐parametric estimator of the characteristic function associated with a copula. It is shown that these statistics behave asymptotically as degenerate V ‐statistics of order four and that the limit distributions have representations in terms of weighted sums of independent chi‐square variables. The consistency of the tests against general alternatives is established and an asymptotically valid parametric bootstrap is suggested for the computation of the critical values of the tests. The behaviour of the new tests in small and moderate sample sizes is investigated with the help of simulations and compared with a competing test based on the empirical copula. Finally, the methodology is illustrated on a five‐dimensional data set.     

Testing for concordance between several criteria

The problem of whether the rankings of some objects given by a set of criteria (or judges) show any agreement or are more or less independent is addressed. The most familiar measure for concordance is the Kendall W coefficient. Classical tests for concordance are the Friedman test and the F test. Legendre [Species associations: the Kendall coefficient of concordance revisited. J. Agric. Biol. Environ. Stat. 2005;10(2):226–245] compared via simulation the Friedman test and its permutation version. Unfortunately, the simulation study of Legendre was very limited because it considered neither the copula aspect nor the F test. Kendall W is a rank-based correlation measure, and therefore it is not affected by the marginal distributions of the underlying variables, but only by the copula of the multivariate distribution. In this article, the simulation study of Legendre is deeply extended by considering the copula aspect as well as the F test. It is shown that the Friedman test is too conservative and less powerful than both the F test and the permutation test for concordance which always have a correct size and behave alike. The F test should be preferred because it is computationally much easier. Surprisingly, the power function of the tests is not much affected by the type of copula.     

A generalization of the bivariate Beta-Binomial distribution

This paper presents a new bivariate discrete distribution that generalizes the bivariate Beta-Binomial distribution. It is generated by Appell hypergeometric function F₁ and can be obtained as a Binomial mixture with an Exton's Generalized Beta distribution. The model has different marginal distributions which are, together with the conditional distributions, more flexible than the Beta-Binomial distribution. It has non-linear regression curves and is useful for random variables with positive correlation. These features make the model very adequate to fit observed data as the two applications included show.     

Modeling Dependence in High Dimensions With Factor Copulas

This article presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high-dimensional applications, involving 50 or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider “scree” plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk. Supplementary materials for this article are available online.     

我们应该选用什么样的相关性指标？

现在大家在实际中使用的、教科书上见到的 ,度量两个变量之间相关性的指标 ,往往都是皮尔逊的相关系数 ,即使在多元分析教材中 ,谈到典型相关系数 ,它实际上也是基于二元的相关系数。用相关系数来反映变量之间的相关性有什么不好呢 ?我们都已用了很长的时间 ,用它说明了许多问题 ,现在为什么要提出选用什么样的相关性指标 ,难道除它之外 ,还有别的相关性指标吗 ?还有哪些可以选用的指标呢 ?凭什么准则来选用什么样的相关性指标呢 ?这些正是我们这篇文章要探讨的问题。金融市场的风险分析这几年有很快的发展 ,原来的一些分析方法很不适应这一…     

Mutual information as a measure of multivariate association: analytical properties and statistical estimation

Mutual information (also known as Kullback–Leibler divergence) can be viewed as a measure of multivariate association in a random vector. The definition incorporates the joint density as well as the marginal densities. We will focus on a representation of mutual information in terms of copula densities that is thus independent of the marginal distributions. This representation yields a different approach to estimating mutual information than the original definition does, as only the copula density has to be estimated. We review analytical properties and examples for selected distributions and discuss methods of nonparametric estimation of copula densities and hence of the mutual information from a sample. Based on a simulation study, we compare the performance of these estimators with respect to bias, standard deviation, and the root mean squared error. The Gauss and the Frank copula are considered as examples.     

Inference for Bivariate Survival Data by Copula Models Adjusted for the Boundary Effect

Copula models describe the dependence structure of two random variables separately from their marginal distributions and hence are particularly useful in studying the association for bivariate survival data. Semiparametric inference for bivariate survival data based on copula models has been studied for various types of data, including complete data, right-censored data, and current status data. This article discusses the boundary effect on these inference procedures, a problem that has been neglected in the previous literature. Specifically, asymptotic distribution of the association estimator on the boundary of parameter space is derived for one-dimensional copula models. The boundary properties are applied to test independence and to study the estimation efficiency. Simulation study is conducted for the bivariate right-censored data and current status data.     

The copula directional dependence by stochastic volatility models

This paper proposes a copula directional dependence by using a bivariate Gaussian copula beta regression with Stochastic Volatility (SV) models for marginal distributions. With the asymmetric copula generated by the composition of two Plackett copulas, we show that our SV copula directional dependence by the Gaussian copula beta regression model is superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of the percent relative efficiency of bias and mean squared error. To validate our proposed method with the real data, we use Brent Crude Daily Price (BRENT), West Texas Intermediate Daily Price (WTI), the Standard & Poor’s 500 (SP) and US 10-Year Treasury Constant Maturity Rate (TCM) so that our copula SV directional dependence is overall superior to the Kim and Hwang (2016) copula directional dependence by an asymmetric GARCH model in terms of precision by the percent relative efficiency of mean squared error. In terms of forecasting using the real financial data, we also show that the Bayesian SV model of the uniform transformed data by a copula conditional distribution yields an improvement on the volatility models such as GARCH and SV.