Fractionally integrated GARCH model with tempered stable distribution: a simulation study

ABSTRACT

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.

KEYWORDS:

• Long memory
• FIGARCH model
• fat-tailed distribution
• tempered stable distribution
• maximum likelihood estimation

AMS SUBJECT CLASSIFICATION:

• C22
• C51

Acknowledgements

We are grateful to the ANU College of Business and Economics and Jiangxi University of Finance and Economics for their financial support. The authors would also like to thank Dave Allen, Felix Chan, Michael McAleer, Paresh Narayan, Morten Nielson, Albert Tsui, Zhaoyong Zhang, participants at the 1st Conference on Recent Developments in Financial Econometrics and Applications, the 8th China R Conference (Nanchang), ANU Research School Brown Bag Seminar, Central University of Finance and Economics Seminar, China Meeting of Econometric Society, Chinese Economists Society China Annual Conference, Econometric Society Australasian Meeting, Ewha Woman's University Department of Economics Seminar, International Congress of the Modeling and Simulation Society of Australia and New Zealand and Shandong University Seminar and three anonymous referees for their helpful comments and suggestions. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In this case, the associated Levy processes are called ‘truncated Levy flights’, the appropriateness of which to be applied in the original GARCH model is also discussed in [14].

2. The calculation of the summation is conducted via fast Fourier transformation algorithm [25]. We thank Jensen and Nielsen for making the computation codes available.

3. Since the mean of ${\epsilon }_t$ is 0, ${\eta }_t$ is sometimes named as standardized residual.

4. The second moment of the Stable distribution only exists when $\alpha =2$. In this case, the symmetric Stable distribution collapses to a Gaussian distribution and cannot describe the fat fails.

5. Specific conditions can be found in [2].

6. As discussed in [33], discrete Fourier transform works most efficiently for N being expressed in terms of a power of 2.

7. The details of which can be found in [31].

8. For distributions with defined density function (like Normal, Student-t and GED), steps (2)–(6) are replaced by generating the density values of ${\eta }_t$ with the estimates of the distribution parameters.

9. The FIGARCH(0,d,0), FIGARCH(1,d,0) and FIGARCH(0,d,1) processes can lead to similar results, which are available upon request.

10. We also considered the cases of 4, 5 and 6 degrees of freedom. The results are robust and available upon request.

11. We also considered the cases of 1.3, 1.5 and 1.8 degrees of freedom. The results are robust and available upon request.

12. We report conditional volatility here as the square root of $h_t$, so that it has the same scale as $r_t$.

Additional information

Funding

This work receives financial support from the ANU College of Business and Economics and Research School of Finance, Actuarial Studies and Applied Statistics.