Truncated and simplified regular vines and their applications

Abstract

In this chapter we present some of the basic concepts that are used throughout the thesis.In particular, we introduce copulas and discuss the important classes of elliptical and Archimedean copulas.Then we present a broad range of bivariate copula families and their relationships to the common dependence measures Kendall's and tail dependence.Bivariate copulas are the building blocks of flexible multivariate copula models, so-called pair copula constructions.Since the number of possible pair copula constructions is quite large, they need to be classified using regular vines which are based on graph theoretical concepts for modeling dependence.Finally, we consider time series models which will be used to analyze and pre-process data in our applications in Chapter 11. CopulasIn the following, we mainly follow Nelsen (2006), while an illustrative introduction to the topic is given in Genest and Favre (2007) and further information can be found, e.g., in Joe (1997).According to Nelsen (2006), copulas can briefly be described as "functions that join or 'couple' multivariate distribution functions to their one-dimensional marginal distribution functions".This description is motivated by the important result of Sklar (1959) and explains the name "copula".It also highlights the main feature of copulas: they allow to model dependency among random variables separately of their margins.We will now formally discuss the concept of copulas and their properties.In doing so we first shortly discuss bivariate copulas by way of illustration and then turn to general multivariate copulas.Definition 2.1 (Bivariate opula.)A two-dimensional copula is a function C : [0, 1] 2 [0, 1] with the following properties:(i) For every u 1 , u 2 [0, 1],C(u 1 , 0) = 0 = C(0, u 2 ) and C(u 1 , 1) = u 1 and C(1, u 2 ) = u 2 .-0.05 0.