Advertisement

Asymptotic properties of the QMLE in a log-linear RealGARCH model with Gaussian errors

  • Caiya Zhang
  • Kaihong Xu
  • Lianfen QianEmail author
Regular Article
  • 92 Downloads

Abstract

To incorporate the realized volatility in stock return, Hansen et al. (J Appl Econ 27:877–906, 2012) proposed a RealGARCH model and conjectured some theoretical properties about the quasi-maximum likelihood estimation (QMLE) for parameters in a log-linear RealGARCH model without rigorous proof. Under Gaussian errors, this paper derives the detailed proof of the theoretical results including consistency and asymptotic normality of the QMLE, hence it solves the conjectures in Hansen et al. (J Appl Econ 27:877–906, 2012).

Keywords

RealGARCH model Quasi-maximum likelihood estimator Consistency and asymptotic normality 

Notes

Acknowledgements

We thank the Editor-in-Chief and the two referees for their helpful comments and suggestions that have led to significant improvements of this paper. Caiya Zhang’s research is partially supported by the Research Projects of Humanities and Social Science of Ministry of Education of China (17YJA910003), Intelligent Plant Factory of Zhejiang Province Engineering Lab. Lianfen Qian’s research is partially supported by the Natural Science Foundation of Zhejiang Province (LY17A010012) and the OURI Curriculum Grant from Florida Atlantic University, USA.

References

  1. Amemiya T (1985) Advanced econometrics. Harvard University Press Cambridge, Cambridge, pp 105–108Google Scholar
  2. Andersen TG, Bollerslev T (1998) Answering the skeptics: yes, standard volatility models do provide accurate forecasts. Int Econ Rev 39:885–905CrossRefGoogle Scholar
  3. Andersen TG, Bollerslev T, Diebold FX, Labys P (2003) Modeling and forecasting realized volatility. Econometrica 71:579–625MathSciNetCrossRefGoogle Scholar
  4. Barndorff-Nielsen OE, Graversen SE, Shephard N (2004) Power variation and stochastic volatility: a review and some new results. J Appl Probab 41A:133–143MathSciNetCrossRefGoogle Scholar
  5. Barndorff-Nielsen OE, Hansen PR, Lunde A (2008) Designing realised kernels to measure the expost variation of equity prices in the presence of noise. Econometrica 76:1481–1536MathSciNetCrossRefGoogle Scholar
  6. Billingsley P (1995) Probability and measures, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  7. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econ 31:307–327MathSciNetCrossRefGoogle Scholar
  8. Brockwell PJ, Davis RA (2009) Time series: theory and methods. Springer, New YorkzbMATHGoogle Scholar
  9. Crosato L, Grossi L (2017) Correcting outliers in GARCH models: a weighted forward approach. Stat Pap.  https://doi.org/10.1007/s00362-017-0903-y
  10. Engle RF (2002) New frontiers of ARCH models. J Appl Econ 17:425–446CrossRefGoogle Scholar
  11. Fan JQ, Qi L, Xiu DC (2014) Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods. J Bus Econ Stat 32:178–191MathSciNetCrossRefGoogle Scholar
  12. Han H, Kristensen D (2014) Asymptotic theory for the QMLE in GARCH-X models with stationary and nonstationary covariates. J Bus Econ Stat 32:416–429MathSciNetCrossRefGoogle Scholar
  13. Hansen PR, Huang Z, Shek H (2012) Realized GARCH: a joint model of returns and realized measures of volatility. J Appl Econ 27:877–906MathSciNetCrossRefGoogle Scholar
  14. Jensen ST, Rahbek A (2004) Asymptotic inference for nonstationary GARCH. Econ Theory 20:1203–1226MathSciNetCrossRefGoogle Scholar
  15. Lee S, Hansen BE (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econ Theory 10:29–52MathSciNetCrossRefGoogle Scholar
  16. Li Q, Zhu F (2017) Mean targeting estimator for the integer-valued GARCH(1, 1) model. Stat Pap.  https://doi.org/10.1007/s00362-017-0958-9
  17. Martens M, Dick D (2007) Measuring volatility with the realized range. J Econ 138:181–207MathSciNetCrossRefGoogle Scholar
  18. Qian LF, Wang SJ (2017) Subject-wise empirical likelihood inference in partial linear models for longitudinal data. Comput Stat Data Anal 111:77–87MathSciNetCrossRefGoogle Scholar
  19. Straumann D, Minkosch T (2006) Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equation approach. Ann Stat 34:2449–2495MathSciNetCrossRefGoogle Scholar
  20. Wang SJ, Qian LF, Carroll RJ (2010) Generalized empirical likelihood methods for analyzing longitudinal data. Biometrika 97(1):79–93MathSciNetCrossRefGoogle Scholar
  21. Zhu F, Wang D (2015) Empirical likelihood for linear and log-linear INGARCH models. J Korean Stat Soc 44(1):150–160MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Computer and Computing ScienceZhejiang University City CollegeHangzhouChina
  2. 2.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations