Characteristic function-based inference for GARCH models with heavy-tailed innovations

ABSTRACT

We consider estimation and goodness-of-fit tests in GARCH models with innovations following a heavy-tailed and possibly asymmetric distribution. Although the method is fairly general and applies to GARCH models with arbitrary innovation distribution, we consider as special instances the stable Paretian, the variance gamma, and the normal inverse Gaussian distribution. Exploiting the simple structure of the characteristic function of these distributions, we propose minimum distance estimation based on the empirical characteristic function of properly standardized GARCH-residuals. The finite-sample results presented facilitate comparison with existing methods, while the new procedures are also applied to real data from the financial market.

KEYWORDS:

• Characteristic function
• GARCH model
• Heavy-tailed distribution
• Minimum distance estimation

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62F10
• Secondary 62M10

Acknowledgment

This work was supported by research grant no. 11699 of the Special Account for Research Grants (ELKE) of the National and Kapodistrian University of Athens.

Notes

1 We use optimization tool constOptim from R-package “stats.”

2 We employ the R-package “tseries.”

3 We use optimization tool nlminb from R-package “stats.”

4 The population mean and variance of each distribution are used to standardize the simulated errors. For details on Hansen's skew Student-t and the generalized hyperbolic skew Student-t see Hansen (1994) and Aas and Haff (2006), respectively. For the simulation of random numbers we used the corresponding R-package routines. The code of A. Patton is adapted in R to generate random numbers from Hansen's skew t distribution, see http://public.econ.duke.edu/ ap172/code.html.

5 In the ML–QML estimation the optimization had problems converging. This was due to the evaluation of the CDF of VG for large values of θ₂. We bounded θ₂ ⩽ 5 to overcome this problem. This was not needed in the CF–QML estimation, but we imposed the bound for comparison.

6 Computer specification: Windows 7 32-bit, Intel Core i3 at 3.2 GHz, RAM 4 GB, single instance of R running at 25% CPU usage.

7 The source of the data is Yahoo Finance.

8 The CF–QML estimation of the Student-t GARCH is as described for the VG and NIG cases with the difference that we consider a grid of integer values 5,6,…,30, in the CF estimation of the degrees of freedom. To evaluate the characteristic function of the Student-t distribution we used the R-package ‘prob.’