Zhu, Ke and Ling, Shiqing
(2013):
*Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models.*
Published in: Annals of Statistics
, Vol. 39, No. 4
(2011): pp. 2131-2163.

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## Abstract

This paper investigates the asymptotic theory of the quasi-maximum exponential likelihood estimators (QMELE) for ARMA–GARCH models. Under only a fractional moment condition, the strong consistency and the asymptotic normality of the global self-weighted QMELE are obtained. Based on this self-weighted QMELE, the local QMELE is showed to be asymptotically normal for the ARMA model with GARCH (finite variance) and IGARCH errors. A formal comparison of two estimators is given for some cases. A simulation study is carried out to assess the performance of these estimators, and a real example on the world crude oil price is given.

Item Type: | MPRA Paper |
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Original Title: | Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models |

English Title: | Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models |

Language: | English |

Keywords: | ARMA–GARCH/IGARCH model; asymptotic normality; global selfweighted/local quasi-maximum exponential likelihood estimator; strong consistency. |

Subjects: | C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General C - Mathematical and Quantitative Methods > C5 - Econometric Modeling |

Item ID: | 51509 |

Depositing User: | Dr. Ke Zhu |

Date Deposited: | 17 Nov 2013 14:33 |

Last Modified: | 29 Sep 2019 06:19 |

References: | BASRAK, B., DAVIS, R. A. and MIKOSCH, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99, 95–115. BERA, A. K. andHIGGINS, M. L. (1993). ARCH models: Properties, estimation and testing. Jounal of Economic Surveys 7 305–366; reprinted in Surveys in Econometrics (L. Oxley et al., eds.) 215–272. Blackwell, Oxford 1995. BERKES, I., HORVÁTH, L. and KOKOSZKA, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9, 201–227. BERKES, I. and HORVÁTH, L. (2004). The efficiency of the estimators of the parameters in GARCH processes. Ann. Statist. 32, 633–655. BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327. BOLLERSLEV, T., CHOU, R. Y. and KRONER, K. F. (1992). ARCH modeling in finance: A review of the theory and empirical evidence. J. Econometrics 52, 5–59. BOLLERSLEV, T., ENGEL, R. F. andNELSON, D. B. (1994). ARCH models. In Handbook of Econometrics, 4 (R. F. Engle and D. L. McFadden, eds.) 2961–3038. North-Holland, Amesterdam. BOUGEROL, P. and PICARD, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. conometrics 52, 115–127. CHAN, N. H. and PENG, L. (2005). Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors. Biometrika 92, 477–484. DAVIS, R. A. and DUNSMUIR, W. T. M. (1997). Least absolute deviation estimation for regression with ARMA errors. J. Theoret. Probab. 10, 481–497. DAVIS, R. A., KNIGHT, K. and LIU, J. (1992). M-estimation for autoregressions with infinite variance. Stochastic Process. Appl. 40, 145–180. ENGLE, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007. FRANCQ, C. and ZAKOÏAN, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10, 605–637. FRANCQ, C. and ZAKOÏAN, J. M. (2010). GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley, Chichester, UK. HALL, P. and YAO, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317. HORVÁTH, L. and LIESE, F. (2004). Lp-estimators in ARCH models. J. Statist. Plann. Inference 119, 277–309. HUBER, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics 221–233. Univ. California Press, Berkeley, CA. KNIGHT, K. (1987). Rate of convergence of centred estimates of autoregressive parameters for infinite variance autoregressions. J. Time Series Anal. 8, 51–60. KNIGHT, K. (1998). Limiting distributions for L1 regression estimators under general conditions. Ann. Statist. 26, 755–770. LANG, W. T., RAHBEK, A. and JENSEN, S. T. (2011). Estimation and asymptotic inference in the first order AR–ARCH model. Econometric Rev. 30, 129–153. LEE, S.-W. and HANSEN, B. E. (1994). Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52. LI, G. and LI, W. K. (2005). Diagnostic checking for time series models with conditional heteroscedasticity estimated by the least absolute deviation approach. Biometrika 92, 691–701. LI, G. and LI, W. K. (2008). Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity. Biometrika 95, 399–414. LING, S. (2005). Self-weighted least absolute deviation estimation for infinite variance autoregressive models. J. R. Stat. Soc. Ser. B Stat. Methodol. 67, 381–393. LING, S. (2007). Self-weighted and local quasi-maximum likelihood estimators for ARMA–GARCH/IGARCH models. J. Econometrics 140, 849–873. LING, S. and LI, W. K. (1997). On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J. Amer. Statist. Assoc. 92, 1184–1194. LING, S. and MCALEER, M. (2003). Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19, 280–310. LUMSDAINE, R. L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1, 1) and covariance stationary GARCH(1, 1) models. Econometrica 64, 575–596. PAN, J.,WANG, H. and YAO, Q. (2007). Weighted least absolute deviations estimation for ARMA models with infinite variance. Econometric Theory 23, 852–879. PENG, L. and YAO, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90, 967–975. POLLARD, D. (1985). New ways to prove central limit theorems. Econometric Theory 1, 295–314. WU, R. and DAVIS, R. A. (2010). Least absolute deviation estimation for general autoregressive moving average time-series models. J. Time Series Anal. 31, 98–112. ZHU, K. (2011). On the LAD estimation and likelihood ratio test in time series. Thesis Dissertion, Hong Kong Univ. Science and Technology. ZHU, K. and LING, S. (2012). The global weighted LAD estimators for finite/infinite variance ARMA(p, q) models. Econometric Theory 28, 1065-1086. ZHU, K. and LING, S. (2013). Quasi-maximum exponential likelihood estimators for a double AR(p) model. Statistica Sinica 23, 251-270. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/51509 |