Elliptical and Skew-Elliptical Regression Models and Their Applications to Financial Data Analytics

Abstract

Various statistical distributions have played significant roles in financial data analytics in recent decades. Among these, elliptical modeling has gained popularity, while the study and application of skew-elliptical modeling have garnered increased attention in various domains. This paper begins by acknowledging the notable accomplishments and contributions of Professor Chris Heyde in the field of financial data modeling. We provide a comprehensive review of elliptical and skew-elliptical modeling, summarizing the latest advancements. In particular, we focus on the characteristics, estimation methods, and diagnostics of elliptical and skew-elliptical distributions in regression and time series models, as well as copula modeling. Furthermore, we discuss several related applications in regression and time series models, including estimation and diagnostic methods. The main objective of this paper is to address the critical need for accurately identifying the underlying elliptical distribution, whether it is elliptical or skew-elliptical. This identification is essential for conducting local influence diagnostics and employing appropriate regression methods using suitable elliptical modeling techniques. To illustrate this process, we present examples that demonstrate the identification of the elliptical distribution, starting with the Box–Jenkins methodology and progressing to copula modeling. The inclusion of copula modeling is motivated by its effectiveness in conjunction with elliptical and skew-elliptical distributions, as it aids in distinguishing between the two. Ultimately, the findings of this paper offer valuable insights, as correctly determining the elliptical and skew-elliptical distribution enables the application of suitable local influence and regression methods, thereby contributing to financial portfolio management, business analytics, and insurance analytics, ensuring the accurate specification of models.