Indirect Estimation of α-Stable Distributions and Processes

Abstract

Summary The α‐stable family of distributions constitutes a generalization of the Gaussian distribution, allowing for asymmetry and thicker tails. Its practical usefulness is coupled with a marked theoretical appeal, as it stems from a generalized version of the central limit theorem in which the assumption of the finiteness of the variance is replaced by a less restrictive assumption concerning a somehow regular behaviour of the tails. Estimation difficulties have however hindered its diffusion among practitioners. Since stably distributed random numbers can be produced thoroughly, we propose an indirect estimation approach which uses a skew‐t distribution as auxiliary model. The properties of this approach are assessed in a detailed simulation study.

Estimating GARCH-type models with symmetric stable innovations: Indirect inference versus maximum likelihood

“…As an alternative to ML, we apply the indirect inference estimation method introduced by Gouriéroux et al (1993), which is a simulation-based technique suitable to solve difficult or intractable estimation problems. This method has already proved to be a valuable candidate for the estimation of the parameters of the stable distribution in Lombardi and Calzolari (2008) and Garcia et al (2011). The idea behind the IndInf estimation method is to replace the model of interest (true model) with an approximated model, which is easier to handle and estimate (auxiliary model).…”

“…In this paper we adapt their approach and integrate the GARCH and TGARCH models in the estimation procedure. As an alternative, we also apply the IndInf method, which proves to be a valuable alternative to estimate the stable parameters (Lombardi and Calzolari, 2008;Garcia et al, 2011). Section 4 gives a thorough description of the two estimation methods and of their practical implementation when estimating the parameters of symmetric stable (T)GARCH models as specified above.…”

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.…”

Do the Innovations in a Monetary VAR Have Finite Variances?

“…Akgiray and Lamoureux (1989), Garcia, Renault, and Veredas (2006), Kogon and Williams (1998), Lombardi andCalzolari (2008), andMcCulloch (1997) Here, we begin with the estimation and diagnostics approach suggested by Nolan (1999Nolan ( , 2001. Three estimates are used here: the quantile method of McCulloch (1986), the characteristic function regression method of Koutrovelis (1980) and Kogon and Williams (1998), and the ML estimate (DuMouchel 1973;Nolan 2001).…”