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Restricted normal mixture QMLE for non-stationary TGARCH(1, 1) models

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Abstract

The threshold GARCH (TGARCH) models have been very useful for analyzing asymmetric volatilities arising from financial time series. Most research on TGARCH has been directed to the stationary case. This paper studies the estimation of non-stationary first order TGARCH models. Restricted normal mixture quasi-maximum likelihood estimation (NM-QMLE) for non-stationary TGARCH models is proposed in the sense that we estimate the other parameters with any fixed location parameter. We show that the proposed estimators (except location parameter) are consistent and asymptotically normal under mild regular conditions. The impact of relative leptokursis and skewness of the innovations’ distribution and quasi-likelihood distributions on the asymptotic efficiency has been discussed. Numerical results lend further support to our theoretical results. Finally, an illustrated real example is presented.

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References

  1. Bai X, Russell J R, Tiao G C. Kurtosis of GARCH and stochastic volatility models with non-normal innovations. J Econometrics, 2003, 114: 349–360

    Article  MATH  MathSciNet  Google Scholar 

  2. Beran R. Minimum Hellinger distance estimates for parametric models. Ann Statist, 1977, 5: 445–463

    Article  MATH  MathSciNet  Google Scholar 

  3. Berkes I, Horváth L. The efficiency of the estimators of the parameters in GARCH processes. Ann Statist, 2004, 32: 633–655

    Article  MATH  MathSciNet  Google Scholar 

  4. Berkes I, Horváth L, Kokoszka P. GARCH processes: Structure and estimation. Bernoulli, 2003, 9: 201–227

    Article  MATH  MathSciNet  Google Scholar 

  5. Bollerslev T. Generalized autoregressive conditional heteroskedasticity. J Econometrics, 1986, 31: 307–327

    Article  MATH  MathSciNet  Google Scholar 

  6. Brown B M. Martingale central limit theorems. Ann Math Statist, 1971, 42: 59–66

    Article  MATH  MathSciNet  Google Scholar 

  7. Chan N H, Ng C T. Statistical inference for non-stationary GARCH(p, q) models. Electron J Statist, 2009, 3: 956–992

    Article  MATH  MathSciNet  Google Scholar 

  8. Engle R F. Autoregressive conditional heteroscedasticity with estimates of variance of U.K. inflation. Econometrica, 1982, 50: 987–1008

    Article  MATH  MathSciNet  Google Scholar 

  9. Engle R F, Rangel J G. The spline GARCH model for unconditional volatility and its global macroeconomic causes. NYU Stern School working paper, 2005

    Google Scholar 

  10. Fan J Q, Qi L, Xiu D C. Non-Gaussian quasi maximum likelihood estimation of GARCH models. http://faculty.chicagobooth.edu/dacheng.xiu/research/NGQMLE.pdf, 2013

    Google Scholar 

  11. Francq C, Zakoïan J M. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 2004, 10: 605–637

    Article  MATH  MathSciNet  Google Scholar 

  12. Francq C, Zakoïan J M. Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models. Econometrica, 2012, 80: 821–861

    Article  MATH  MathSciNet  Google Scholar 

  13. Francq C, Zakoïan J M. Inference in non stationary asymmetric GARCH models. Ann Statist, 2013, 41: 70–98

    Article  Google Scholar 

  14. Glosten L R, Jaganathan R, Runkle D. On the relation between the expected values and the volatility of the nominal excess return on stocks. J Finance, 1993, 48: 1779–1801

    Article  Google Scholar 

  15. González-Rivera G, Drost F C. Efficiency comparisons of maximum-likelihood-based estimators in GARCH models. J Econometrics, 1999, 93: 93–111.

    Article  MATH  MathSciNet  Google Scholar 

  16. Hall P, Yao Q. Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica, 2003, 71: 285–317

    Article  MATH  MathSciNet  Google Scholar 

  17. Hsieh D A. Testing nonliear dependence in daily foreign exchange rates. J Business, 1989, 62: 339–368

    Article  Google Scholar 

  18. Hwang S Y, Baek J S, Park J A, et al. Explosive volatilities for threshold-GARCH processes generated by asymmetric innovations. Statist probab Lett, 2010, 80: 26–33

    Article  MATH  MathSciNet  Google Scholar 

  19. Jensen S T, Rahbek A. Asymptotic normality of the QMLE estimator of ARCH in the nonstatinary case. Econometrica, 2004, 72: 641–646

    Article  MATH  MathSciNet  Google Scholar 

  20. Jensen S T, Rahbek A. Asymptotic inference for nonstationary GARCH. Econometric Theory, 2004, 20: 1203–1226

    Article  MATH  MathSciNet  Google Scholar 

  21. Lee S W, Hansen B E. Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory, 1994, 10: 29–52

    Article  MATH  MathSciNet  Google Scholar 

  22. Lee T W, Lee S. Normal mixture quasi-maximum likelihood estimator for GARCH models. Scand J Statist, 2009, 36: 157–170

    MATH  Google Scholar 

  23. Li C W, Li W K. On a double-threshold autoregressive heteroscedastic time series model. J Appl Econometrics, 1996, 11: 253–274

    Article  Google Scholar 

  24. Linton O, Pan J Z, Wang H. Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory, 2010, 26: 1–28

    Article  MATH  MathSciNet  Google Scholar 

  25. Loretan M, Phillips P C B. Testing the covariance stationarity of heavy-tailed time series. J Empir Finance, 1994, 1: 211–248

    Article  Google Scholar 

  26. Lumsdaine R L. Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1, 1) models. Econometrica, 1996, 64: 575–596

    Article  MATH  MathSciNet  Google Scholar 

  27. Mikosch T, Stărică C. Limit theory for the sample autocorrelations and extremes of a GARCH(1, 1) process. Ann Statist, 2000, 28: 1427–1451

    Article  MATH  MathSciNet  Google Scholar 

  28. Pan J Z, Wang H, Tong H. Estimation and tests for power-transformed and threshold GARCH models. J Econometrics, 2008, 142: 352–378

    Article  MathSciNet  Google Scholar 

  29. Stărică C, Herzel S, Nord T. Why Does the GARCH(1, 1) Model Fail to Provide Sensible Longer-Horizon Volatility Forecasts? Chalmers: Chalmers University of Technology, 2005

    Google Scholar 

  30. White H. Maximum likelihood estimation of misspecified models. Econometrica, 1982, 20: 1–25

    Article  Google Scholar 

  31. Zhang Z, Li W K, Yuen K C. On a mixture GARCH time-series model. J Time Ser Anal, 2006, 27: 577–597

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Finance, Central University of Finance and Economics, Beijing, 100081, China

    Hui Wang

  2. Department of Mathematics and Statistics, University of Strathclyde, Glasgow, G11XH, UK

    JiaZhu Pan

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  1. Hui Wang
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  2. JiaZhu Pan
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Correspondence to Hui Wang.

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Wang, H., Pan, J. Restricted normal mixture QMLE for non-stationary TGARCH(1, 1) models. Sci. China Math. 57, 1341–1360 (2014). https://doi.org/10.1007/s11425-014-4815-1

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  • DOI: https://doi.org/10.1007/s11425-014-4815-1

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