Simulation-based Estimation Methods for α-Stable Distributions and Processes

The α-stable family of distributions constitutes a generalization of theGaussian distribution, allowing for asymmetry and thicker tails. Its practicalusefulness is coupled with a marked theoretical appeal, given that it stemsfrom a generalized version of the central limit theorem in which the assump-tion of the finiteness of the variance is replaced by a much less restrictive oneconcerning a somehow regular behavior of the tails. Estimation difficultieshave however hindered its diffusion among practitioners.Since simulated values from α-stable distributions can be straightfor-wardly obtained, the indirect inference approach could prove useful to over-come these estimation difficulties. In this paper I will provide a description ofhow to implement such a method by using the skew-t distribution of Azzalini& Capitanio (2003) as an auxiliary model. The indirect inference approachwill be introduced in the setting of the estimation of the distribution parame-ters and then extended to linear time series models with stable disturbances.The performance of this estimation method is then assessed on simulateddata. An application on time-series models for the inflation rate concludesthe paper.