A simulation study on the distributions of disturbances in the GARCH model

Generalized autoregressive conditional heteroskedastic (GARCH) model is a standard approach to study the volatility behaviour of financial time series. The original specification of GARCH model is developed based on Normal distribution for the disturbances, which cannot accommodate fat-tailed properties commonly existing in financial time series. Consequently, the resulting estimates are not efficient. Traditionally, the Student's t-distribution and General Error Distribution (GED) are used alternatively to solve this problem. However, a recent study points out that those alternative distributions lack stability under aggregation. This leaves the appropriate choice of the distribution of disturbances in the GARCH model still an open question. In this paper, we present the theoretical features and desirability of the tempered stable distribution. Further, we conduct a series of simulation studies to demonstrate that the GARCH model with this distribution consistently outperforms those with the Normal, Student-t and GED distributions. This result is robust with empirical evidence of the S&P 500 daily return. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the financial volatility in general contexts with a GARCH-type specification.