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

Financial returns exhibit conditional heteroscedasticity, asymmetric responses of their volatility to negative and positive returns (leverage effects) and fat tails. The alpha-stable distribution is a natural candidate for capturing the tail-thickness of the conditional distribution of financial returns, while the GARCH-type models are very popular in depicting the conditional heteroscedasticity and leverage effects. However, practical implementation of alpha-stable distribution in finance applications has been limited by its estimation difficulties. The performance of the indirect inference approach using GARCH models with Student's t distributed errors as auxiliary models is compared to the maximum likelihood approach for estimating GARCH-type models with symmetric alpha-stable innovations. It is shown that the expected efficiency gains of the maximum likelihood approach come at high computational costs compared to the indirect inference method.