Evaluating and improving GARCH-based volatility forecasts with range-based estimators

Abstract

This article investigates the feasibility of using range-based estimators to evaluate and improve Generalized Autoregressive Conditional Heteroscedasticity (GARCH)-based volatility forecasts due to their computational simplicity and readily availability. The empirical results show that daily range-based estimators are sound alternatives for true volatility proxies when using Superior Predictive Ability (SPA) test of Hansen (2005) to assess GARCH-based volatility forecasts. In addition, the inclusion of the range-based estimator of Garman and Klass (1980) can significantly improve the forecasting performance of GARCH-t model.

Keywords:

• range-based estimators
• GARCH-based volatility forecasts
• SPA test

JEL Classification::

• C52
• C53

Acknowledgements

The authors are grateful to the editor (Dr Mark P. Taylor) and two anonymous referees for their valuable comments and suggestions that have significantly improved the article. Jui-Cheng Hung also acknowledges the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC98-2410-H-262-010. All remaining errors are the authors’ responsibility.

Notes

¹ See Andersen and Bollerslev (1997, 1998) and McMillan and Speight (2004) for detailed discussions.

² The RV is severely biased if based on intraday returns that are sampled at a high frequency, due to microstructure noise, see, e.g. Andreou and Ghysels (2002) and Zhang et al. (2005), and see Hansen and Lunde (2004, 2006a) for empirical studies that characterize the properties of market microstructure noise.

³ Similarly, Andersen et al. (1999) also observed a considerable improvement in the out-of-sample forecasting performance of the GARCH model. Martens (2001) compared both GARCH-based methods for two exchanges rates, and found that the most accurate intraday-GARCH model, which proved to be the model with the highest sampling frequency, could not outperform the daily GARCH model extended with intraday volatility. These studies therefore indicate that intraday return series contain incremental information for longer-run volatility forecasts when used in combination with GARCH models.

⁴ Notably, as the absolute values of forecast errors are less than unity, taking their square root will place a heavier weighting on the under-predictions. If the absolute value of all forecast errors were greater than unity, the MME(U) would need to square the errors in order to achieve the desired penalty (Brailsford and Faff, 1996).

⁵ The period of VIX data in this article starts from 2 January 2003 to 30 December 2005. We thank professor Shu-fang Yuan for sharing these data.

⁶ The p-values of conservative test for SPA test is not reported because it is quite sensitive to the inclusion of poor and irrelevant models in the comparison, while the consistent (SPA_c) and the liberal test (SPA_l) are not. The program code of SPA test can be downloaded from the website of Peter Reinhard Hansen.

⁷ Wilhelmsson (2006) investigated the predictive ability of the GARCH(1,1) model with various error distributions, and found that allowing for a leptokurtic error distribution is demonstrated to significantly improve variance forecasts for alternative forecast horizons. For our research purpose, the Student t distribution is incorporated with the GARCH model to control the possible improvement of using alternative error distribution assumption.

⁸ For the sake of achieving the desired penalty, the forecasted volatilities and the true volatility proxy are both divided by 10 to make the forecast errors less than unity.

⁹ While neglecting the possible improvement of using alternative error distribution assumption, we find that the GARCH-VIX improves the forecasting performance of the GARCH model when evaluating with MAE, MME(U) and MME(O) loss functions. To save space, the results of GARCH model incorporated with RV, VIX, PK, GK and RS are not reported here but available upon request.