Construction

Hierarchical models offer a practical solution to overcome the limitations of Archimedean copulas when dealing with a larger number of variables, allowing for flexible dependence modeling by employing specific inner copulas to model groups of variables within an outer dependence structure. This hierarchical construction, represented as a tree or dendrogram, captures the hierarchical relationships between variables and increases the strength of dependence as the groups become more nested in the structure. In Sect. 3.1, we explore the motivation for hierarchical structures in copulas, focusing on the trivariate and multivariate cases of HACs. We discuss their flexibility in modeling dependence, illustrate the pattern for constructing more complex HACs in the trivariate case, and introduce the concept of undirected trees to represent the hierarchical structure. Furthermore, we provide a general definition of hierarchical Archimedean copulas. Section 3.4 introduces the nesting condition for HACs, which involves checking the first derivative of the composition of two generators. Alternative approaches, such as using outer generators and Lévy subordinators, are mentioned. We present different analytical conditions on inner and outer generators for compatibility in nesting ACs, categorized based on the type of generators used. We also highlight that completely monotone pairs of generators are not necessary and weaker conditions can be used. In Sect. 3.6, we discuss identifiability, stating that the parameter of an HAC is not identifiable due to scaling of the generator functions. However, additional constraints can be imposed to ensure identifiability. Section 3.7 briefly addresses how binary structures of HACs relate to non-binary ones, noting that binary structures simplify structure and parameter estimation. In order to identify an equivalent binary structure for a non-binary HAC, it is suggested to replace higher-order ACs with binary counterparts. Lastly, we introduce hierarchical outer power ACs (HOPACs), which allow for nested ACs that can be outpower transformed. We present the sufficient condition for constructing a parametric HOPAC and emphasize their relationship to hierarchical Archimax copulas.