Computational Economics

, Volume 37, Issue 3, pp 301–330 | Cite as

Volatility Modeling by Asymmetrical Quadratic Effect with Diminishing Marginal Impact

  • Alex YiHou Huang


This study presents evidence of an asymmetrical quadratic effect from financial asset return on volatility. The relationships between the two variables are quadratic for both positive and negative returns and systematically different in the two regimes. The convex relations are observed showing that extreme shocks have a diminishing marginal impact on volatility. A threshold quadratic model under GARCH framework is developed to capture the effect and applied to major stock indices. The empirical outcomes of quadratic regressions and in-sample estimations significantly confirm the asymmetrical quadratic behavior. With application of S&P500 series, both diagnoses of in-sample estimations and evaluations of out-of-sample forecasts verify the proposed specification as a valid alternative volatility modeling.


Volatility modeling Quadratic Asymmetric Threshold GARCH 

JEL Classification

C22 C53 G10 G17 


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  1. Andersen T. G. (1994) Stochastic autoregressive volatility: A framework for volatility modeling. Mathematical Finance 4: 75–102Google Scholar
  2. Baillie R. T., Bollerslev T., Mikkelsen H. O. (1996) Fractionally integrated generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 74: 3–30CrossRefGoogle Scholar
  3. Bakshi G., Ju N., Ou-Yang H. (2006) Estimation of continuous-time models with an application to equity volatility dynamics. Journal of Financial Economics 82: 227–249CrossRefGoogle Scholar
  4. Bauwens L., Laurent S., Rombouts J. (2006) Multivariate GARCH models: A survey. Journal of Applied Econometrics 21: 79–109CrossRefGoogle Scholar
  5. Bekaert G., Wu G. (2000) Asymmetric volatility and risk in equity markets. Review of Financial Studies 13: 1–42CrossRefGoogle Scholar
  6. Berkes I., Horvath L., Kokoszka P. (2003) GARCH processes: Structure and estimation. Bernoulli 9: 201–227CrossRefGoogle Scholar
  7. Blair B., Poon S., Taylor S. J. (2001) Forecasting S&P100 volatility: The incremental information content of implied volatilities and high frequency index returns. Journal of Econometrics 105: 5–26CrossRefGoogle Scholar
  8. Bollerslev T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31: 307–328CrossRefGoogle Scholar
  9. Bollerslev T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69: 542–547CrossRefGoogle Scholar
  10. Bollerslev T., Ghysels E. (1996) Periodic autoregressive conditional heteroscedasticity. Journal of Business and Economic Statistics 14: 139–151CrossRefGoogle Scholar
  11. Bollerslev T., Chou R. Y., Kroner K. F. (1992) ARCH modeling in finance: A review of the theory and empirical evidences. Journal of Econometrics 52: 5–59CrossRefGoogle Scholar
  12. Carnero M. A., Peña D., Ruiz E. (2007) Effects of outliers on the identification and estimation of the GARCH models. Journal of Time Series Analysis 28: 471–497CrossRefGoogle Scholar
  13. Charles A. (2008) Forecasting volatility with outliers in GARCH models. Journal of Forecasting 27: 551–565CrossRefGoogle Scholar
  14. Chiarella C., To T.-D. (2006) The multifactor nature of the volatility of futures markets. Computational Economics 27: 163–183CrossRefGoogle Scholar
  15. Enders W. (2004) Applied econometric time series. Wiley Series in Probability and Statistics. Wiley, New York, NYGoogle Scholar
  16. Engle R. F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50: 987–1008CrossRefGoogle Scholar
  17. Engle R. F., Bollerslev T. (1986) Modelling the persistence of conditional variance. Econometric Review 5: 1–50CrossRefGoogle Scholar
  18. Engle R. F., Ng V. K. (1993) Measuring and testing the impact of news on volatility. Journal of Finance 48: 1749–1778CrossRefGoogle Scholar
  19. Eraker B., Johannes M., Polson N. (2003) The impact of jumps in volatility and returns. Journal of Finance 58: 1269–1300CrossRefGoogle Scholar
  20. Fama E. F. (1965) The behavior of stock market prices. Journal of Business 38: 34–105CrossRefGoogle Scholar
  21. Francq C., Zakoian J. M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10: 605–637CrossRefGoogle Scholar
  22. Franses P. H., van Dijk D. (1996) Forecasting stock market volatility using (non-linear) GARCH models. Journal of Forecasting 15: 229–235CrossRefGoogle Scholar
  23. Gilli M., Kellezi E. (2006) An application of extreme value theory for measuring financial risk. Computational Economics 27: 207–228CrossRefGoogle Scholar
  24. Glosten L. R., Jagannathan R., Runkle D. E. (1993) On the relation between the expected value and the volatility of the normal excess return on stocks. Journal of Finance 48: 1779–1801CrossRefGoogle Scholar
  25. Haas M., Mittnik S., Paolella M. S. (2004) A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2: 493–530CrossRefGoogle Scholar
  26. Hsieh D. (1989) Modeling heteroskedasticity in daily foreign exchange rates. Journal of Business and Economic Statistics 7: 307–317CrossRefGoogle Scholar
  27. Laurent S. (2004) Analytical derivatives of the APARCH model. Computational Economics 24: 51–57CrossRefGoogle Scholar
  28. Ling S., McAleer M. (2002) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106: 109–117CrossRefGoogle Scholar
  29. Lopez J. (2001) Evaluating the predictive accuracy of volatility models. Journal of Forecasting 20: 87–109CrossRefGoogle Scholar
  30. Nelson D. B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59: 347–370CrossRefGoogle Scholar
  31. Rabemananjara R., Zakoian J. M. (1993) Threshold ARCH models and asymmetries in volatility. Journal of Applied Econometrics 8: 31–49CrossRefGoogle Scholar
  32. Ramsey J. B. (1969) Tests for specification errors in classical linear least-squares analysis. Journal of the Royal Statistical Association (Series B) 71: 350–371Google Scholar
  33. Sentana E. (1995) Quadratic ARCH models. Review of Economic Studies 62: 639–661CrossRefGoogle Scholar
  34. Thiel H. (1966) Applied economic forecasting. Rand McNally, Chicago, ILGoogle Scholar
  35. van Dijk D., Franses P. H., Lucas A. (2002) Testing for ARCH in the presence of additive outliers. Journal of Applied Econometrics 14: 539–562CrossRefGoogle Scholar
  36. Yu J. (2005) On leverage in a stochastic volatility model. Journal of Econometrics 127: 165–178CrossRefGoogle Scholar
  37. Zakoian J. M. (1994) Threshold heteroscedastic models. Journal of Economic Dynamics and Control 18: 931–995CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of FinanceYuan Ze UniversityChung-LiTaiwan

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