Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Improving forecasting performance using covariate-dependent copula models

Copulas provide an attractive approach to the construction of multivariate distributions with flexible marginal distributions and different forms of dependences. Of particular importance in many areas is the possibility of forecasting the tail-dependences explicitly. Most of the available approaches are only able to estimate tail-dependences and correlations via nuisance parameters, and cannot be used for either interpretation or forecasting. We propose a general Bayesian approach for modeling and forecasting tail-dependences and correlations as explicit functions of covariates, with the aim of improving the copula forecasting performance. The proposed covariate-dependent copula model also allows for Bayesian variable selection from among the covariates of the marginal models, as well as the copula density. The copulas that we study include the Joe-Clayton copula, the Clayton copula, the Gumbel copula and the Student’s $t$-copula. Posterior inference is carried out using an efficient MCMC simulation method. Our approach is applied to both simulated data and the S&P 100 and S&P 600 stock indices. The forecasting performance of the proposed approach is compared with those of other modeling strategies based on log predictive scores. A value-at-risk evaluation is also performed for the model comparisons.     

Dynamic stochastic copula models: estimation,inference and applications

We propose a new dynamic copula model in which the parameter characterizing dependence follows an autoregressive process. As this model class includes the Gaussian copula with stochastic correlation process, it can be viewed as a generalization of multivariate stochastic volatility models. Despite the complexity of the model, the decoupling of marginals and dependence parameters facilitates estimation. We propose estimation in two steps, where first the parameters of the marginal distributions are estimated, and then those of the copula. Parameters of the latent processes (volatilities and dependence) are estimated using efficient importance sampling. We discuss goodness‐of‐fit tests and ways to forecast the dependence parameter. For two bivariate stock index series, we show that the proposed model outperforms standard competing models. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust portfolio optimization

We address the problem of estimating risk-minimizing portfolios from a sample of historical returns, when the underlying distribution that generates returns exhibits departures from the standard Gaussian assumption. Specifically, we examine how the underlying estimation problem is influenced by marginal heavy tails, as modeled by the univariate Student-t distribution, and multivariate tail-dependence, as modeled by the copula of a multivariate Student-t distribution. We show that when such departures from normality are present, robust alternatives to the classical variance portfolio estimator have lower risk.     

Modeling Financial Sector Joint Tail Risk in the Euro Area

We develop a novel high‐dimensional non‐Gaussian modeling framework to infer measures of conditional and joint default risk for numerous financial sector firms. The model is based on a dynamic generalized hyperbolic skewed‐t block equicorrelation copula with time‐varying volatility and dependence parameters that naturally accommodates asymmetries and heavy tails, as well as nonlinear and time‐varying default dependence. We apply a conditional law of large numbers in this setting to define joint and conditional risk measures that can be evaluated quickly and reliably. We apply the modeling framework to assess the joint risk from multiple defaults in the euro area during the 2008–2012 financial and sovereign debt crisis. We document unprecedented tail risks between 2011 and 2012, as well as their steep decline following subsequent policy actions. Copyright © 2016 John Wiley & Sons, Ltd.     

Time series copulas for heteroskedastic data

We propose parametric copulas that capture serial dependence in stationary heteroskedastic time series. We suggest copulas for first‐order Markov series, and then extend them to higher orders and multivariate series. We derive the copula of a volatility proxy, based on which we propose new measures of volatility dependence, including co‐movement and spillover in multivariate series. In general, these depend upon the marginal distributions of the series. Using exchange rate returns, we show that the resulting copula models can capture their marginal distributions more accurately than univariate and multivariate generalized autoregressive conditional heteroskedasticity models, and produce more accurate value‐at‐risk forecasts.     

Modeling non-normal corporate bond yield spreads by copula

This research focuses on modeling for how corporate bond yield spreads are affected by explanatory variables such as equity volatility, interest rate volatility, r, slope, rating, liquidity, coupon rate, and maturity. The existing literature assumes normality and linearity in the analysis, which is not the case in our sample. Thus, through a powerful and flexible copula approach, we study the dependence at the mean of the joint distribution by using the Gaussian copula marginal regression method and the dependence structure at the tails by using various copula functions. To our knowledge, this is the first application of the copula marginal regression model to bond market data. In addition, we employ several copula functions to test for the tail dependence between yield spreads and other explanatory variables. We find stronger tail dependence in the joint upper tail for the relation between equity volatility and yield spreads, among others. This result indicates the positive effect of equity volatility on yield spreads in the upper tail is greater than that in the low tail. This finding should be useful to practitioners, such as investors. By relying on better-fitting, more meaningful statistical models, this paper contributes to the extant literature on how corporate bond yield spreads are determined.     

A Skew‐normal copula‐driven GLMM

This paper presents a method for fitting a copula‐driven generalized linear mixed models. For added flexibility, the skew‐normal copula is adopted for fitting. The correlation matrix of the skew‐normal copula is used to capture the dependence structure within units, while the fixed and random effects coefficients are estimated through the mean of the copula. For estimation, a Monte Carlo expectation–maximization algorithm is developed. Simulations are shown alongside a real data example from the Framingham Heart Study.     

Estimation of multivariate models for time series of possibly different lengths

We consider the problem of estimating parametric multivariate density models when unequal amounts of data are available on each variable. We focus in particular on the case that the unknown parameter vector may be partitioned into elements relating only to a marginal distribution and elements relating to the copula. In such a case we propose using a multi‐stage maximum likelihood estimator (MSMLE) based on all available data rather than the usual one‐stage maximum likelihood estimator (1SMLE) based only on the overlapping data. We provide conditions under which the MSMLE is not less asymptotically efficient than the 1SMLE, and we examine the small sample efficiency of the estimators via simulations. The analysis in this paper is motivated by a model of the joint distribution of daily Japanese yen–US dollar and euro–US dollar exchange rates. We find significant evidence of time variation in the conditional copula of these exchange rates, and evidence of greater dependence during extreme events than under the normal distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

An inflated multivariate integer count hurdle model: an application to bid and ask quote dynamics

In this paper we develop a model for the conditional inflated multivariate density of integer count variables with domain ?ⁿ, n ∈ ?. Our modelling framework is based on a copula approach and can be used for a broad set of applications where the primary characteristics of the data are: (i) discrete domain; (ii) the tendency to cluster at certain outcome values; and (iii) contemporaneous dependence. These kinds of properties can be found for high‐ or ultra‐high‐frequency data describing the trading process on financial markets. We present a straightforward sampling method for such an inflated multivariate density through the application of an independence Metropolis–Hastings sampling algorithm. We demonstrate the power of our approach by modelling the conditional bivariate density of bid and ask quote changes in a high‐frequency setup. We show how to derive the implied conditional discrete density of the bid–ask spread, taking quote clusterings (at multiples of 5 ticks) into account. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of copula-based semiparametric time series models

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric marginal distributions and parametric copula functions, while the copulas capture all the scale-free temporal dependence of the processes. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. These results are used to obtain root-n consistent and asymptotically normal estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles. The semiparametric conditional quantile estimators are automatically monotonic across quantiles, which is attractive for portfolio conditional value-at-risk calculations.     

Simulating non‐normal distributions with specified L‐moments and L‐correlations

This paper derives a procedure for simulating continuous non‐normal distributions with specified L‐moments and L‐correlations in the context of power method polynomials of order three. It is demonstrated that the proposed procedure has computational advantages over the traditional product‐moment procedure in terms of solving for intermediate correlations. Simulation results also demonstrate that the proposed L‐moment‐based procedure is an attractive alternative to the traditional procedure when distributions with more severe departures from normality are considered. Specifically, estimates of L‐skew and L‐kurtosis are superior to the conventional estimates of skew and kurtosis in terms of both relative bias and relative standard error. Further, the L‐correlation also demonstrated to be less biased and more stable than the Pearson correlation. It is also shown how the proposed L‐moment‐based procedure can be extended to the larger class of power method distributions associated with polynomials of order five.     

Financial dependence analysis: applications of vine copulas

This paper features the application of a novel and recently developed method of statistical and mathematical analysis to the assessment of financial risk, namely regular vine copulas. Dependence modelling using copulas is a popular tool in financial applications but is usually applied to pairs of securities. Vine copulas offer greater flexibility and permit the modelling of complex dependence patterns using the rich variety of bivariate copulas that can be arranged and analysed in a tree structure to facilitate the analysis of multiple dependencies. We apply regular vine copula analysis to a sample of stocks comprising the Dow Jones index to assess their interdependencies and to assess how their correlations change in different economic circumstances using three different sample periods around Global Financial Crisis (GFC).: pre‐GFC (January 2005 to July 2007), GFC (July 2007 to September 2009) and post‐GFC periods (September 2009 to December 2011). The empirical results suggest that the dependencies change in a complex manner, and there is evidence of greater reliance on the Student‐t copula in the copula choice within the tree structures for the GFC period, which is consistent with the existence of larger tails in the distributions of returns for this period. One of the attractions of this approach to risk modelling is the flexibility in the choice of distributions used to model co‐dependencies. The practical application of regular vine metrics is demonstrated via an example of the calculation of the Value at Risk of a portfolio of stocks.     

A Unifying Framework for Marginalised Random‐Intercept Models of Correlated Binary Outcomes

We demonstrate that many current approaches for marginal modelling of correlated binary outcomes produce likelihoods that are equivalent to the copula‐based models herein. These general copula models of underlying latent threshold random variables yield likelihood‐based models for marginal fixed effects estimation and interpretation in the analysis of correlated binary data with exchangeable correlation structures. Moreover, we propose a nomenclature and set of model relationships that substantially elucidates the complex area of marginalised random‐intercept models for binary data. A diverse collection of didactic mathematical and numerical examples are given to illustrate concepts. Copyright © 2013 John Wiley & Sons, Ltd.     

HEGY Tests in the Presence of Moving Averages*

We analyze the asymptotic distributions associated with the seasonal unit root tests of Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average process. Although both the t‐ and F‐type tests suffer from scale and shift effects compared with the presumed null distributions when a fixed order of autoregressive augmentation is applied, these effects disappear when the order of augmentation is sufficiently large. However, as found by Burridge and Taylor (2001) for the autoregressive case, individual t‐ratio tests at the semi‐annual frequency are not pivotal even with high orders of augmentation, although the corresponding joint F‐type statistic is pivotal. Monte Carlo simulations verify the importance of the order of augmentation for finite samples generated by seasonally integrated moving average processes.     

Robust Likelihood Methods Based on the Skew-t and Related Distributions

The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew‐t distribution is explored in more detail and reasons are given to adopt this option as a sensible general‐purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide‐ranging examples with real data are provided in support of the claim.     

Keynesian impulses versus Solow residuals: identifying sources of business cycle fluctuations

We employ a neoclassical business‐cycle model to study two sources of business‐cycle fluctuations: marginal efficiency of investment shocks, and total factor productivity shocks. The parameters of the model are estimated using a Bayesian procedure that accommodates prior uncertainty about their magnitudes; from these estimates, posterior distributions of the two shocks are obtained. The postwar US experience suggests that both shocks are important in understanding fluctuations, but that total factor productivity shocks are primarily responsible for beginning and ending recessions. Copyright © 2000 John Wiley & Sons, Ltd.     

The t Copula and Related Copulas

The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.     

Flexible Estimation of Copulas: An Application to the US Housing Crisis

Zimmer (‘The role of copulas in the housing crisis’, Review of Economics and Statistics 2012; 94 : 607–620) provides an interesting case study of the pitfalls of using parametric copulas to understand the US housing crisis in the latter part of 2000s. The original study by Zimmer (2012) employs a finite‐mixture copula to illustrate that the symmetry of the Gaussian copula may not be tenable, especially for US housing price data during the time period from 1975:Q2 to 2009:Q1. We undertake a replication of his study in a wide sense. First, we replicate the study by incorporating revised data and then extending the dataset to include the most recent data. Second, we implement a nonparametric copula estimator recently proposed by Racine (‘Mixed data kernel copulas’, Empirical Economics forthcoming) to the parametrically filtered data used in Zimmer (2012). Our replication finds that the application of the nonparametric copula to the same and extended filtered data provides an alternative flexible specification for copulas. However, the overall cautionary message of the flexible‐form copula espoused in Zimmer (2012) remains. Copyright © 2015 John Wiley & Sons, Ltd.