Linear factor copula models and their properties

Abstract

We consider a special case of factor copula models with additive common factors and independent components.These models are flexible and parsimonious with O(d) parameters where d is the dimension. The linear structure allows one to obtain closed form expressions for some copulas and their extreme-value limits. These copulas can be used to model data with strong tail dependencies, such as extreme data. We study the dependence properties of these linear factor copula models and derive the corresponding limiting extreme-value copulas with a factor structure. We show how parameter estimates can be obtained for these copulas and apply one of these copulas to analyse a financial data set. ), in which the dependence structure is selected based on the likelihood, the models with a factor structure can be nicely interpreted. One example is a credit portfolio when some severe economic shocks can affect all the portfolio components, leading to multiple defaults. These shocks usually cannot be easily measured, and there might be many different factors contributing to these shocks, so it is natural to assume that these are unobserved variables.In this paper, we study the special class of factor copula models with linear structures, as proposed by Krupskii & Joe (2013). These copulas can handle a wide range of dependencies and have O(d) parameters. To construct a linear factor copula, we use a linear combination of common factors and independent factors, each having the same distribution. Similar ideas were used in the construction of generalized Archimedean copulas (Rogge & Schönbucher, 2003) and models based on comonotonic factors (Hua & Joe, 2017). Linear factor copula models are a special case of the latter models with linear loadings. The limiting extreme-value copulas (Gudendorf & Segers, 2010) with factor structures can be derived for this class of models, in closed form in some cases, and the parameters can be efficiently estimated using a composite maximum likelihood approach (for continuous copulas) or the method of moments (for copulas with singular components).The rest of the paper is organized as follows. In Section 2, we introduce the class of one-factor copula models with linear structures and study their dependence properties. We derive the limiting extreme-value copulas for this class of models and further extend this approach to models with p ≥ 2 factors in Section 3. As a special case of this class of linear factor copula models, we introduce an extension of the Marshall-Olkin copula (Marshall & Olkin, 1967) with p ≥ 2 factors affecting all of the components in a system. In Section 4, we show how the copula parameter estimates can be obtained for the models introduced in the two previous sections. In Section 5, we apply one of the proposed linear factor copula models to a financial data set, and Section 6 concludes with a discussion.

Modeling spatial tail dependence with Cauchy convolution processes

“…, Z d ) as m → ∞, where "→ d " denotes convergence in distribution. It can be seen that ( 15) is a linear factor model with m 2 independent and identically distributed common factors W k,l as considered in Krupskii and Genton [42]. Let F j,m be the CDF of Z j,m , j = 1, .…”

Why Ordinal Variables Can (Almost) Always Be Treated as Continuous Variables: Clarifying Assumptions of Robust Continuous and Ordinal Factor Analysis Estimation Methods

“…If the ordinal nature of variables would be taken seriously, more sophisticated modeling strategies for factor analysis that try to estimate more flexible distributions are required (Jin and Yang-Wallentin, 2017;Foldnes and Grønneberg, 2019). A particularly attractive distribution class is the factor copula model (Krupskii and Joe, 2013;Nikoloulopoulos and Joe, 2015;Ackerer and Vatter, 2017;Krupskii and Genton, 2018). Copula models decompose a joint distribution for modeling into marginal distributions and modeling the dependency structure.…”

Why Ordinal Variables Can (Almost) Always be Treated as Continuous Variables: Clarifying Assumptions of Robust Continuous and Ordinal Factor Analysis Estimation Methods

“…If the ordinal nature of variables would be taken seriously, more sophisticated modeling strategies for factor analysis that try to estimate more flexible distributions are required (Foldnes & Grønneberg, 2019;Jin & Yang-Wallentin, 2017). A particularly attractive distribution class would be factor copula models (Ackerer & Vatter, 2017;Krupskii & Genton, 2018;Krupskii & Joe, 2013;Nikoloulopoulos & Joe, 2015). Copula models decompose a joint distribution for modeling into marginal distributions and modeling the dependency structure.…”