Indirect estimation of elliptical stable distributions

Abstract

We present an indirect estimation approach for elliptical stable distributions which relies on the use of a multivariate t distribution as auxiliary model. This distribution is also elliptical and we show that its parameters have a one-to-one relationship with those of the elliptical stable, therefore making the proposed indirect approach especially suitable. Standard asymptotic properties are also shown and we analyze the finite sample behavior of the estimators via a comprehensive Monte Carlo study. An application to 27 emerging markets stock indexes concludes the paper.

On estimating the tail index and the spectral measure of multivariate $$\alpha $$ α -stable distributions

“…Several researchers have shown that some multivariate financial data can be fitted by multivariate α-stable distributions, see Nolan (2003), Lombardi andVeredas (2009), Kabašinskas et al (2009), and Dominicy and Veredas (2013). Therefore, the estimation of multivariate α-stable distributions is an important issue.…”

“…Using spherical harmonic analysis, estimation of the spectral measure was investigated in Pivato and Seco (2003). Indirect estimation of an elliptical α-stable distribution was considered in Lombardi and Veredas (2009). Ogata (2013) proposed using the generalized empirical likelihood (GEL) method with estimating function constructed by empirical and theoretical characteristic functions.…”

Testing conditional asymmetry: A residual-based approach

“…Hence z n t $ SKSTð0; 1,x,nÞ and y t 9O tÀ1 $ SKSTðm t ,s 2 t ,x,nÞ. The skewed-t density has been used a distribution for a GARCH model (Lambert and Laurent, 2001b,a), for computation of Value at Risk (Giot and Laurent, 2003), indirect estimation of stable densities (Lombardi and Calzolari, 2008;Garcia et al, 2011;Lombardi and Veredas, 2009) and its multivariate extension has been proposed by Bauwens and Laurent (2005). This is a flexible distribution with four parameters that accounts for the four main features of a distribution: location, scale, asymmetry and tail thickness.…”

Stable Mixture GARCH Models

“…While of value for a small number of assets, those models will not be practical for even modest portfolios, let alone large ones. In addition, attempting to extend that model to support the multivariate stable distribution is not trivial (but see Lombardi and Veredas, 2009;Bonato, 2011; and the references therein).…”