Hierarchical Archimedean Copulas

In today’s data-driven world, the analysis of multiple variables and their joint behavior has become increasingly important. Copulas offer a versatile and powerful framework for understanding and modeling the dependence structure between variables. This chapter serves as a guide to the concept of copula, highlighting its main properties and key aspects discussed in each section. In Sect. 1.1, we delve into the motivation behind studying copulas and their broad range of applications in various fields. We recognize the significant contribution of Abe Sklar to copula theory, emphasizing his pivotal role in introducing copulas to the research community. Section 1.2 then focuses on the fundamental concepts of copula theory, starting with their definition as bivariate distribution functions with uniform margins. The properties of copulas, such as the rectangular inequality and the boundary conditions, are explored. Sklar’s theorem, which establishes the relationship between a bivariate distribution and its copula, is introduced as a key result. In Sect. 1.3, we discuss the simplest copulas and their properties. We explore the concept of elliptical copulas, which are a special class of copulas based on elliptical distributions. Additionally, we delve into different measures of dependence used to quantify the strength of the relationship between variables. Section 1.6 expands the discussion from bivariate to multivariate copulas. We explore the extension of copulas to handle dependence among multiple variables, allowing for a more comprehensive analysis of complex probabilistic models.

In Sect. 2.2, we delve into the Archimedean property and its connection to triangular norms. We discuss how bivariate Archimedean copulas are linked to a special class of triangular norms used to represent logical conjunctions.

Furthermore, in Sect. 2.3, we describe parametric families of Archimedean copulas. We refer to the works of Nelsen (2006) and Joe (2014) for a comprehensive list of popular generators and their properties. We highlight the Gumbel, Clayton, Frank, and Joe copulas, discussing their respective generator functions, inverse functions, copula formulas, and their relations to Kendall’s \(\tau \). We also provide visualizations of the density functions of these copulas for various marginal distributions.

Lastly, we introduce the concept of outer power transformation, which allows for the modification of existing copula families. By fixing a copula family and applying the outer power transformation, we can adjust the properties of the resulting copulas. This provides additional flexibility, which is particularly useful for tail dependence modeling.

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.

HACs provide a powerful and systematic approach for capturing complex dependence patterns by decomposing the joint distribution into multiple levels of dependence. This unique characteristic empowers researchers and practitioners to gain deeper insights into the relationships among variables, making HACs valuable tools for various applications such as risk management, portfolio optimization, and simulation studies.

In this chapter, we explore the fundamental properties of HACs and shed light on their key characteristics. First, in Sect. 4.1, we delve into an important relationship between an HAC and its bivariate margins, which lays the foundation for understanding higher dimensional structures. Next, we investigate a novel method for uniquely decomposing an HAC structure into a set of trivariate HAC structures in Sect. 4.2. It is shown that this decomposition can be exploited for HAC estimation.

Moreover, we highlight that the entire structure of an HAC can be recovered solely from its matrix of pairwise Kendall’s \(\tau \) in Sect. 4.3. This finding demonstrates the richness of the information contained in the Kendall’s \(\tau \) matrix and its practical implications for HAC analysis.

Finally, we discuss other essential probabilistic and statistical features for practitioners in Sect. 4.4. These include the probabilistic ordering of the HAC, its significance in extreme value theory, and methods to condense the information of the multivariate HAC-distributed variables into a single dimension.

In numerous applications, copulas are widely recognized for their ability to capture complex dependencies. However, the use of copulas can pose a significant challenge due to the difficulty in accessing the related distributions and densities analytically. In such scenarios, the Monte Carlo approach provides the only viable option to obtain an approximation of the distribution of interest. Consequently, for a specific class of copulas, having a computationally efficient sampling algorithm is critical to ensure practicality and accuracy.

In this chapter, we introduce a few sampling algorithms for copulas. The first algorithm, presented in Sect. 5.1, is a general approach that can be applied to any copula, but it may become computationally infeasible in high-dimensional cases. To address this limitation, we present in Sect. 5.2 a second algorithm that is specifically tailored for exchangeable ACs and is more efficient in higher dimensions. Furthermore, we demonstrate how this second algorithm can be extended to handle HACs with arbitrary structures.

Such an extension is presented in Sect. 5.3, where the sampling procedure assumes complete monotonicity of all generators appearing in a HAC, as described in Sect. 2.​1. While this assumption is commonly used in applications involving HACs, as discussed in Chap. 7, it is not strictly necessary for constructing a HAC, as outlined in Sect. 3.​4. As such, alternative sampling algorithms that do not rely on complete monotonicity are also available, and interested readers can refer to Mai (2019) for examples and a comparison with the algorithms presented below. Additionally, for readers seeking more general sampling methods for various types of copulas, we highly recommend the comprehensive review by Mai and Scherer (2012b).

Finally, Sect. 5.4 presents a sampling algorithm for HACs based on outer power transformations of AC generators, discussed in Sect. 2.​4. It includes an example involving a 6-variate copula to illustrate its application.

In typical settings, a copula model for a random vector \(X = (X_1,\dots , X_d)\) arises when the copula \(C(\cdot )\) of X is unknown but assumed to belong to a class where \(\mathcal {O}\) is an open subset of \(\mathbb {R}^p\) for some integer \(p \geq 1\). When \(C_\theta (\cdot )\) is supposed to be a d-HAC \(C_{(\mathcal {V},\mathcal {E},\Psi )}\) in the sense of Definition 3.​2, the parameter (vector) \(\theta \) could be represented by the triplet \((\mathcal {V},\mathcal {E},\Psi )\), which fully determines the possible HAC model. In practice, this model typically contains the following three ingredients:

It is necessary to realize that all three ingredients are interconnected. Without knowing the Archimedean family, the parameters cannot be estimated, and without knowing the structure, the model cannot even be defined and thus neither the family nor the parameters can be obtained. This chapter attempts to present an answer to the following question: Under the assumption where \(\mathcal {C}_0\) is a set of d-HACs parametrized by those three ingredients, how these parameters could be estimated given a random sample \((x_{11},\dots ,x_{1d}), \dots , (x_{n1},\dots , x_{nd})\) drawn from \((X_1,\dots , X_d)\)?

In the literature, we often find approaches that focus just on a single ingredient. These then either assume the remaining ingredients to be known a priori (for example, using the dominant or expert knowledge) or circumvent their necessity. For example, Segers and Uyttendaele (2014), Matsypura et al. (2016), and Uyttendaele (2018) focus purely on the structure ingredient, whereas Savu and Trede (2010) focus on the parameter ingredient. Knowledge acquired by these researchers opens important doors for further investigations: In the case of Savu and Trede (2010), we obtain the best possible estimators in the sense of maximum likelihood. Thus, no other parameter estimator can be better, in terms of consistency and efficiency, in fixed samples while simultaneously estimating structure and parameters. Using knowledge from Segers and Uyttendaele (2014), Matsypura et al. (2016), and Uyttendaele (2018), we obtain the structure that can be used in estimating parameters following Savu and Trede (2010). However, this two-step procedure contains some pitfalls since errors made on one of the steps are propagated to the other.

Another branch of approaches tries to estimate all three ingredients at once (estimation of the structure and parameters with the selection of the AC family), see Górecki et al. (2017b), or simplify the task by fixing one family for all nested ACs and estimate the remaining two ingredients, see Okhrin et al. (2013b), Górecki et al. (2016, 2021), and Cossette et al. (2019).

In the rest of this chapter, we recall these approaches applied to each of the ingredients one by one. Before we do so, we briefly describe a general framework for all estimation approaches.

Modeling temporal dependency holds significant importance across diverse domains like finance, economics, and resource allocation. By capturing the intricate relationships between past and future observations, these models not only enhance prediction accuracy but also play a crucial role in effective risk management and optimal resource allocation. In finance, understanding temporal dependencies is essential for making informed investment decisions and quantifying market risks, while in economics, it aids in designing policies that consider the impact of historical factors on future outcomes.

Over time, the HAC has gained attraction in economics and finance, particularly for its efficacy in handling clustered variables and capturing implied correlations. Its prowess in modeling Value-at-Risk across Basel regulatory phases and its role in understanding collateralized debt obligations during the 2008 financial crisis have bolstered its significance. The copula’s reputation, previously challenged by media, has been reinstated through subsequent research. Current financial trends, like high-frequency trading, have opened avenues for accurate joint distribution modeling of daily returns using HAC. These three financial applications are discussed in this chapter.

While running these empirical studies, researchers have noticed that contrary to other models, in HAC not only the parameters but also the structure are changing. This observation led to another series of papers investigating exactly this issue. This chapter discusses in detail two temporal HAC models and several applications of the HAC.

In this chapter, we explore three software toolboxes that enable stochastic dependence modeling with HACs across different programming languages. Among these, the R package copula available at https://CRAN.R-project.org/package=copula stands out as a comprehensive package for copula modeling, which has been extensively discussed in Yan (2007) and Kojadinovic and Yan (2010). After it incorporated the nacopula package (Hofert & Mächler, 2011), which delivers functionalities for sampling from HACs, it supports dependence modeling also with HACs. Another R package for practitioners interested in HAC modeling is the HAC package available at https://CRAN.R-project.org/package=HAC, discussed in Okhrin and Ristig (2014, 2020). Both these packages are described below in Sect. 8.1. For those preferring MATLAB or Octave, the HACopula toolbox available at https://github.com/gorecki/HACopula (discussed in Górecki et al., 2020) could be an ideal choice. This toolbox is described in Sect. 8.2. While we do not intend to cover all the possible capabilities of these toolboxes, we will present the key concepts, enabling readers to take their first steps in HAC modeling with their preferred software and interpret the results effectively.