With the growing availability of high-frequency data, long memory has become a popular topic in finance research. Fractionally Integrated GARCH (FIGARCH) model is a standard approach to study the long memory of financial volatility. The original specification of FIGARCH model is developed using Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Traditionally, the Student-t distribution and General Error Distribution (GED) are used instead to solve that problem. However, a recent study points out that the Student-t lacks stability. Instead, the Stable distribution is introduced. The issue of this distribution is that its second moment does not exist. To overcome this new problem, the tempered stable distribution, which retains most attractive characteristics of the Stable distribution and has defined moments, is a natural candidate. In this paper, we describe the estimation procedure of the FIGARCH model with tempered stable distribution and conduct a series of simulation studies to demonstrate that it consistently outperforms FIGARCH models with the Normal, Student-t and GED distributions. An empirical evidence of the S&P 500 hourly return is also provided with robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the high-frequency financial volatility in general contexts with a FIGARCH-type specification.
- fat-tailed distribution
- FIGARCH model
- Long memory
- maximum likelihood estimation
- tempered stable distribution