Francq, Christian and Zakoian, Jean-Michel
(2021):
*Testing the existence of moments and estimating the tail index of augmented garch processes.*

Preview |
PDF
MPRA_paper_110511.pdf Download (1MB) | Preview |

## Abstract

We investigate the problem of testing finiteness of moments for a class of semi-parametric augmented GARCH models encompassing most commonly used specifications. The existence of positive-power moments of the strictly stationary solution is characterized through the Moment Generating Function (MGF) of the model, defined as the MGF of the logarithm of the random autoregressive coefficient in the volatility dynamics. We establish the asymptotic distribution of the empirical MGF, from which tests of moments are deduced. Alternative tests relying on the estimation of the Maximal Moment Exponent (MME) are studied. Power comparisons based on local alternatives and the Bahadur approach are proposed. We provide an illustration on real financial data, showing that semi-parametric estimation of the MME offers an interesting alternative to Hill's nonparametric estimator of the tail index.

Item Type: | MPRA Paper |
---|---|

Original Title: | Testing the existence of moments and estimating the tail index of augmented garch processes |

Language: | English |

Keywords: | APARCH model; Bahadur slopes; Hill's estimator; Local asymptotic power; Maximal moment exponent; Moment generating function |

Subjects: | C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C58 - Financial Econometrics |

Item ID: | 110511 |

Depositing User: | Christian Francq |

Date Deposited: | 07 Nov 2021 21:58 |

Last Modified: | 07 Nov 2021 21:58 |

References: | Aue A., Berkes, I. and L. Horv\'ath (2006) Strong approximation for the sums of squares of augmented GARCH sequences. Bernoulli 12 583--608. Baek, C., Pipiras, V., Wendt, H. and P. Abry (2009) Second order properties of distribution tails and estimation of tail exponents in random difference equations. Extremes 12 361--400. Basrak, B., Davis, R.A. and T. Mikosch (2002) Regular variation of GARCH processes. Stochastic Processes and their Applications 99 95--116. Berkes, I., Horv\'ath, L. and P.S. Kokoszka (2003) Estimation of the Maximal Moment Exponent of a GARCH(1,1) sequence. Econometric Theory 19 565--586. Berkes, I. and L. Horv\'ath (2004) The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics 32 633--655. Billingsley, P. (1961) The Lindeberg-L\'evy theorem for martingales. Proceedings of the American Mathematical Society 12 788--792. Blasques, F., Gorgi, P., Koopman, S. J., and O. Wintenberger (2018). Feasible invertibility conditions and maximum likelihood estimation for observation-driven models. Electronic Journal of Statistics 12 1019--1052. Chan, N.H., Li, D., Peng, L. and R. Zhang (2013) Tail index of an AR(1) model with ARCH(1) errors. Econometric Theory 29 920--940. Creal, D., Koopman, S. J., and A. Lucas (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics 28 777--795. Davis, R. and T. Mikosch (2009). Extreme value theory for GARCH processes. In T. Andersen, R. Davis, J.-P. Kreiss, and T. Mikosch (Eds.), Handbook of Financial Time Series, 187--200. New York: Springer. Delaigle, A., Meister, A., and J. Rombouts (2016). Root-T consistent density estimation in GARCH models. Journal of Econometrics 192 55--63. Ding, Z., Granger, C. W., and R.F. Engle (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance 1 83--106. Drees, H. (2000) Weighted approximations of tail processes for $\beta$-mixing random variables. Annals of Applied Probability 10 1274--1301. Drees, H., Resnick, S., and L. de Haan (2000) How to make a Hill plot. The Annals of Statistics 28 254--274. Drost, F. C. and C. A. J. Klaassen (1997) Efficient estimation in semiparametric GARCH models. Journal of Econometrics 81 193--221. Drost, F.C., Klaassen, C.A.J. and B.J.M. Werker (1997) Adaptive estimation in time-series models. Annals of Statistics 25 786--817. Francq, C. and J-M. Zako\"{i}an (2013a) Inference in nonstationary asymmetric GARCH models. The Annals of Statistics 41 1970--1998. Francq, C. and J-M. Zako\"{i}an (2013b) Optimal predictions of powers of conditionally heteroskedastic processes. Journal of the Royal Statistical Society - Series B 75 345--367. Francq, C. and J-M. Zako\"{i}an (2021a) Testing the existence of moments for GARCH processes. Journal of Econometrics, forthcoming. Francq, C. and J-M. Zako\"{i}an (2021b) LAN of GARCH-type and score-driven time-series models. Discussion paper. Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. The Annals of Applied Probability 1 126--166. Hamadeh, T.. and J-M. Zako\"{i}an (2011) Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH Processes. Journal of Statistical Planning and Inference 141 488--507. Harvey, A.(2013) Dynamic models for volatility and heavy tails. Cambridge University Press. He, C. and T. Ter{\"a}svirta (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92 173--192. Heinemann, A. (2019) A bootstrap test for the existence of moments for GARCH processes. Preprint arXiv:1902.01808v3. Hill, J.B. (2015) Tail index estimation for a filtered dependent time series. Statistica Sinica 25, 609--629. Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131 207--248. Lee, S. and M. Taniguchi (2005) Asymptotic theory for ARCH-SM models: LAN and residual empirical processes. Statistica Sinica 15 215--234. Ling, S. and M. McAleer (2002) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106 109--117. Ling, S. and M. McAleer (2003) Adaptative estimation in nonstationary ARMA models with GARCH errors. The Annals of Statistics 31 642--674. Mikosch, T. and C. St\u{a}ric\u{a} (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. The Annals of Statistics 28 1427--1451. Resnick, S.I. and C. St\u{a}ric\u{a} (1998) Tail index estimation for dependent data. Annals of Applied Probability 8 1156--1183. Straumann, D. (2005) Estimation in conditionally heteroscedastic time series models. Lecture Notes in Statistics, Springer Berlin Heidelberg. Straumann, D., and T. Mikosch (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. The Annals of Statistics 34 2449--2495. Trapani, L. (2016) Testing for (in)finite moments. Journal of Econometrics 191, 57--68. Zhang, R., Li, C., and L. Peng (2019) Inference for the tail index of a GARCH(1,1) model and an AR(1) model with ARCH(1) errors. Econometric Reviews 38 151--169. Zhang, R. and S. Ling (2015) Asymptotic inference for AR models with heavy-tailed G-GARCH noises. Econometric Theory 31 880--890. Zhu, K. and S. Ling (2011) Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH/IGARCH models. The Annals of Statistics 39 2131--2163. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/110511 |