Maximum Likelihood Estimation for Conditionally Heteroscedastic Models when the Innovation Process is in the Domain of Attraction of a Stable Law

TL;DR

This paper establishes the theoretical properties of maximum likelihood estimators for conditionally heteroscedastic models with stable law innovations, relevant for modeling heavy-tailed financial data.

Contribution

It proves consistency and asymptotic normality of MLEs under stable law assumptions and extends results to convergence in distribution, with applications to financial risk modeling.

Findings

MLEs are strongly consistent for stable innovations

Asymptotic normality of estimators is established

Application to stable Value-at-Risk in finance

Abstract

We prove the strong consistency and the asymptotic normality of the maximum likelihood estimator of the parameters of a general conditionally heteroscedastic model with $\alpha$-stable innovations. Then, we relax the assumptions and only suppose that the innovation process converges in distribution toward a stable process. Using a pseudo maximum likelihood estimator with a stable density, we also obtain the strong consistency and the asymptotic normality of the estimator. This framework seems relevant for financial data exhibiting heavy tails. We apply this method to several financial index and compute stable Value-at-Risk.