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On IGARCH and convergence of the QMLE for misspecified GARCH models

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Author Info

  • Anders Tolver Jensen

    ()
    (Department of Natural Sciences, University of Copenhagen)

  • Theis Lange

    ()
    (Department of Economics, University of Copenhagen & CREATES)

Abstract

We address the IGARCH puzzle by which we understand the fact that a GARCH(1,1) model tted by quasi maximum likelihood estimation to virtually any financial dataset exhibit the property that alpha^hat + beta^hat is close to one. We prove that if data is generated by certain types of continuous time stochastic volatility models, but fitted to a GARCH(1,1) model one gets that alpha^hat + beta^hat tends to one in probability as the sampling frequency is increased. Hence, the paper suggests that the IGARCH effect could be caused by misspecification. The result establishes that the stochastic sequence of QMLEs do indeed behave as the deterministic parameters considered in the literature on filtering based on misspecified ARCH models, see e.g. Nelson (1992). An included study of simulations and empirical high frequency data is found to be in very good accordance with the mathematical results.

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Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2009-06.

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Length: 33
Date of creation: 19 Feb 2009
Date of revision:
Handle: RePEc:aah:create:2009-06

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Web page: http://www.econ.au.dk/afn/

Related research

Keywords: GARCH; Integrated GARCH; Misspecification; High frequency exchange rates;

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References

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  1. Nelson, Daniel B & Foster, Dean P, 1994. "Asymptotic Filtering Theory for Univariate ARCH Models," Econometrica, Econometric Society, Econometric Society, vol. 62(1), pages 1-41, January.
  2. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, Econometric Society, vol. 50(4), pages 987-1007, July.
  3. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, Elsevier, vol. 4(2-3), pages 115-158, June.
  4. R. F. Engle & A. J. Patton, 2001. "What good is a volatility model?," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 1(2), pages 237-245.
  5. Nelson, Daniel B. & Foster, Dean P., 1995. "Filtering and forecasting with misspecified ARCH models II : Making the right forecast with the wrong model," Journal of Econometrics, Elsevier, Elsevier, vol. 67(2), pages 303-335, June.
  6. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, Econometric Society, vol. 61(4), pages 909-27, July.
  7. Francq, Christian & Zako an, Jean-Michel, 2000. "Estimating Weak Garch Representations," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 16(05), pages 692-728, October.
  8. Ding, Zhuanxin & Granger, Clive W. J., 1996. "Modeling volatility persistence of speculative returns: A new approach," Journal of Econometrics, Elsevier, Elsevier, vol. 73(1), pages 185-215, July.
  9. Bollerslev, Tim & Engle, Robert F, 1993. "Common Persistence in Conditional Variances," Econometrica, Econometric Society, Econometric Society, vol. 61(1), pages 167-86, January.
  10. Werker, B.J.M. & Drost, F.C., 1996. "Closing the GARCH gap: Continuous time GARCH modeling," Open Access publications from Tilburg University, Tilburg University urn:nbn:nl:ui:12-72561, Tilburg University.
  11. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, Econometric Society, vol. 71(2), pages 579-625, March.
  12. Nelson, Daniel B., 1992. "Filtering and forecasting with misspecified ARCH models I : Getting the right variance with the wrong model," Journal of Econometrics, Elsevier, Elsevier, vol. 52(1-2), pages 61-90.
  13. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2007. "Models for S&P500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices," CREATES Research Papers, School of Economics and Management, University of Aarhus 2007-37, School of Economics and Management, University of Aarhus.
  14. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, Elsevier, vol. 74(1), pages 3-30, September.
  15. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  16. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, Elsevier, vol. 129(1-2), pages 121-138.
  17. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
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