We introduce a new type of heavy-tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), to the GARCH and Glosten-Jagannathan-Runkle (1993) GARCH models, and compare its empirical performance with two other popular types of heavy-tailed distribution, the Student’s t distribution and the normal inverse Gaussian distribution (NIG), using a variety of asset return series. Our results illustrate that there is no overwhelmingly dominant distribution in fitting the data under the GARCH framework, although the NRIG distribution performs slightly better than the other two types of distribution. For market indexes series, it is important to introduce both GJR-terms and the NRIG distribution to improve the models’ performance, but it is ambiguous for individual stock prices series. Our results also show the GJR-GARCH NRIG model has practical advantages in quantitative risk management. Finally, the convergence of numerical solutions in maximum-likelihood estimation of GARCH and GJR-GARCH models with the three types of heavy-tailed distribution is investigated.
Heavy-tailed distribution GARCH model Model comparison Numerical solution