Indirect estimation of GARCH models with alpha-stable innovations

Several studies have highlighted the fact that heavy-tailedness of asset returns can be the consequence of conditional heteroskedasticity. GARCH models have thus become very popular, given their ability to account for volatility clustering and, implicitly, heavy tails. However, these models encounter some difficulties in handling financial time series, as they respond equally to positive and negative shocks and their tail behavior remains too short even with Student-t error terms. To overcome these weaknesses we apply GARCH-type models with alpha-stable innovations. The stable family of distributions constitutes a generalization of the Gaussian distribution that has intriguing theoretical and practical properties. Indeed it is stable under addiction and, having four parameters, it allows for asymmetry and heavy tails. Unfortunately stable models do not have closed likelihood function, but since simulated values from α-stable distributions can be straightforwardly obtained, the indirect inference approach is particularly suited to the situation at hand. In this work we provide a description of how to estimate a GARCH(1,1) and a TGARCH(1,1) with symmetric stable shocks using as auxiliary model a GARCH(1,1) with skew-t innovations. Monte Carlo simulations, conducted using GAUSS, are presented and finally the proposed models are used to estimate the IBM weekly return series as an illustration of how they perform on real data.