Vinzenz Erhardt & Claudia Czado
We suggest an approximative method for sampling high-dimensional count random variables with a specified Pearson correlation. As in the continuous case copulas can be used to construct multivariate discrete distributions. Here we utilize Gaussian copulas for the construction. A major task, however, is to determine the appropriate copula parameters to obtain the specified target correlation. Very often, the fact that for the Gaussian copula the correlation matrix of the multivariate normal distribution is not equal to the correlation of the sampled (discrete) outcomes, is simply neglected.
We will introduce an optimization routine to determine the copula parameters sequentially using bisection in order to determining correlations and partial correlation parameters of the copula. Therefore, we need to break our T-dimensional copula down to a decomposition of bivariate copulas with only one parameter each. For this, we use canonical vines, a graphical tool to organize such pair-copula decompositions of high-dimensional distributions. We will illustrate that our sampling approach generates very accurate results even in high dimensions in several settings and outperforms the widely used 'naive' sampling approach.
Identification of conditional independence in multivariate copula data is, in general, a diffcult problem. Confidence intervals for parameters of interest in the maximum likelihood approach are not always available. We tackle this problem in a Bayesian framework. First I briefly outline a Bayesian estimation of pair-copula constructions based on bivariate t-copulas. Then I present our model selection procedure for identifying conditional independence in multivariate copula data. It is based on a reversible jump MCMC of Peter Green (Biometrika 82, pp. 711-732). Finally I illustrate applications of our algorithm to Norwegian returns and Euro swap rates data. This is a joint work with Claudia Czado.
In order to capture the dependency among exchange rates we construct semiparametric multivariate copula models with ARMA-GARCH margins. As multivariate copula models we utilize pair-copula constructions (PCC) such as regular and canonical vines. As building block of the PCC's we use bivariate t-copulas for different tail dependence between pairs of exchange rates. Alternatively we also consider a non Gaussian directed acylic graph (DAG) model which can be imbedded as a special PCC. We apply these models to Euro exchange rates. A nonnested model comparison technique is developed to compare DAG, regular and canonical vine based models. This is joint work with Aleksey Min, Tanja Baumann and Rada Dakovic.
Fat-tail distributions are common place, particularly in insurance and finance. This phenomenon is well documented and can be analyzed by standard statistical tools. Extreme-value dependence occurs just as frequently, e.g., between claim data or financial assets. Yet it is seldom recognized and accounted for. This survey talk will describe how to test, measure and account for extreme dependence from a copula modeling perspective.
Consider d-dimensional regular vines. For d=3, 4, 5, there are respectively one, two and six equivalence classes of regular vines. Some comparisons of different vine copulas can be made from how they are simulated. Results on dependence comparisons of the different classes of vine copulas can help in the choice of vine for statistical modeling. For example, we compare the different vine copulas when the level k linking copulas are all set at C_k, k=1,...,d-1.
A multivariate data set, which exhibit complex patterns of dependence, particularly in the tails, can be modelled using a cascade of lower-dimensional copulae. In this paper, we compare two such models that differ in their representation of the dependency structure, namely the nested Archimedean construction (NAC) and the pair-copula construction (PCC). The NAC is much more restrictive than the PCC in two respects. There are strong limitations on the degree of dependence in each level of the NAC, and all the bivariate copulas in this construction has to be Archimedean. Based on an empirical study with two different four-dimensional data sets; precipitation values and equity returns, we show that the PCC provides a better fit than the NAC and that it is computationally more efficient. Hence, we claim that the PCC is more suitable than the NAC for hich dimensional modelling.
The arguments leading to the classical AIC does not hold for the case of parametric copulae models when using the pseudo maximum likelihood procedure. We derive a proper correction, and name it the Copula Information Criterion (CIC). The CIC is, as the copula, invariant to the underlying marginal distributions, but is fundamentally different from the AIC formula erroneously used by practitioners. We further observe that bias-correctionÐbased information criteria do not exist for a large class of commonly used copulae, such as the Gumbel or Joe copulae. This motivates the investigation of median unbiased information criteria in the fully parametric setting, which then could be ported over to the current, more complex, semi-parametric setting.
We propose a new dynamic model for volatility and dependence in high dimensions, that allows for departures from the normal distribution, both in the marginals and in the dependence. For the marginal distributions, we use non-Gaussian GARCH models, that are designed to capture skewness and kurtosis. The dependence is modeled with a dynamic canonical vine copula, which can be decomposed into a cascade of bivariate conditional copulas. Due to this decomposition, the model does not suffer from the curse of dimensionality. The canonical vine autoregressive (CAVA) captures asymmetries in the dependence structure. The model is applied to 95 SP500 stocks. By conditioning on the market index and on sector indexes, the dependence structure is much simplified and can be considered as a non-linear version of the CAPM or of a market model with sector effects. The model is shown to deliver highly accurate estimates of Value-at-Risk.
Archimedean copulas constitute an important class of dependence functions which enjoy considerable popularity in a number of practical applications. This fact is mainly due to their simple algebraic form which permits tractable investigation of the dependence characteristics and opens possibilities for feasible statistical inference. So far however, most of the work on Archimedean copulas has been restricted to either the bivariate case or the situation where the generator is a Laplace transform of some non-negative random variable. In this talk, we focus on Archimedean copulas in high dimensions whose generators are arbitrary and provide necessary and sufficient conditions for their existence. We will also show that Archimedean copulas are naturally linked to simplex distributions. This allows for generalizations of numerous results from the bivariate case to any dimension, for easy construction of new families and for a feasible simulation procedure which overcomes the tedious inversion of a Laplace transform. Finally, implications for extreme-value dependence will be explored, along with generalizations to asymmetric copula families.
This paper proposes a new algorithm of generating a regular vine. We start building the vine from the top node and progress to the lower trees ensuring that the regularity condition is satisfied. We suggest one strategy of choosing the 'most suitable' vine for the correlation matrix. The 'best vine' is the one which nodes of top trees correspond to the smallest absolute values of partial correlations. If we assume that we can assign the independent copula to nodes of the vine with small absolute value of partial correlation then this algorithm can be used as a preprocessing step of fitting a vine to high dimensional data which reduces the number of conditional copulas that have to be fitted.