Robust and efficient estimation of GARCH models based on Hellinger distance

Abstract

It is well known that financial data frequently contain outlying observations. Almost all methods and techniques used to estimate GARCH models are likelihood-based and thus generally non-robust against outliers. Minimum distance method, as an important tool for statistical inferences and a competitive alternative for achieving robustness, has surprisingly not been well explored for GARCH models. In this paper, we proposed a minimum Hellinger distance estimator (MHDE) and a minimum profile Hellinger distance estimator (MPHDE), depending on whether the innovation distribution is specified or not, for estimating the parameters in GARCH models. The construction and investigation of the two estimators are quite involved due to the non-i.i.d. nature of data. We proved that the MHDE is a consistent estimator and derived its bias in explicit expression. For both of the proposed estimators, we demonstrated their finite-sample performance through simulation studies and compared with the well-established methods including MLE, Gaussian Quasi-MLE, Non-Gaussian Quasi-MLE and Least Absolute Deviation estimator. Our numerical results showed that MHDE and MPHDE have much better performance than MLE-based methods when data are contaminated while simultaneously they are very competitive when data is clean, which testified to the robustness and efficiency of the two proposed MHD-type estimations.

Keywords:

• GARCH models
• kernel estimation
• maximum likelihood estimation
• minimum (profile) Hellinger distance estimation
• robustness

AMS 2010 subject classifications:

• Primary 62F10
• 62G05
• secondary 62G07
• 62G35

Acknowledgments

The authors thank the Editor, the Associate Editor and three referees for their helpful comments and suggestions that have led to significant improvements in the manuscript. The authors acknowledge with gratitude the support for this research via Discovery Grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada [grant number RGPIN-2018-04328].