Electronic Journal of Statistics

Statistical inference for non-stationary GARCH(p,q) models

Ngai Hang Chan and Chi Tim Ng

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Abstract

This paper studies the quasi-maximum likelihood estimator (QMLE) of non-stationary GARCH(p,q) models. By expressing GARCH models in matrix form, the log-likelihood function is written in terms of the product of random matrices. Oseledec’s multiplicative ergodic theorem is then used to establish the asymptotic properties of the log-likelihood function and thereby, showing the weak consistency and the asymptotic normality of the QMLE for non-stationary GARCH(p,q) models.

Article information

Source
Electron. J. Statist., Volume 3 (2009), 956-992.

Dates
First available in Project Euclid: 17 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1253195941

Digital Object Identifier
doi:10.1214/09-EJS452

Mathematical Reviews number (MathSciNet)
MR2540848

Zentralblatt MATH identifier
1326.62184

Subjects
Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
Asymptotic normality consistency non-stationary GARCH model Oseledec’s multiplicative ergodic theorem product of random matrices quasi-maximum likelihood estimator

Citation

Chan, Ngai Hang; Ng, Chi Tim. Statistical inference for non-stationary GARCH( p , q ) models. Electron. J. Statist. 3 (2009), 956--992. doi:10.1214/09-EJS452. https://projecteuclid.org/euclid.ejs/1253195941


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References

  • [1] Arnold, L., Crauel, H. and Eckmann, J.P. (1991)., Lyapunov exponents: Proceedings of a conference held in Oberwolfach, May 28-June 2, 1990. Lecture Notes in Mathematics 1486. Springer-Verlag, Berlin.
    Zentralblatt MATH: 0733.00015
  • [2] Berkes, L. and Horváth, L. (2004). The efficiency of the estimators of the parameters in GARCH processes., Annals of Statistics 32, 633–655.
    Zentralblatt MATH: 1048.62082
    Digital Object Identifier: doi:10.1214/009053604000000120
    Project Euclid: euclid.aos/1083178941
  • [3] Berkes, L., Horváth, L. and Kokoszka, P. (2003). GARCH processes: structure and estimation., Bernoulli 9, 201–228.
  • [4] Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series., Journal of Econometrics 52, 115–127.
    Zentralblatt MATH: 0746.62087
    Digital Object Identifier: doi:10.1016/0304-4076(92)90067-2
  • [5] Cohen, J.E., Kesten, H. and Newman, C.M. (1986a)., Random Matrices and Their Applications. American Mathematical Society, Rhode Island.
  • [6] Cohen, J.E., Kesten, H. and Newman, C.M. (1986b). Oseledec’s multiplicative ergodic theorem: a proof, in Random matrices and their applications, edited by Cohen, J.E., Kesten, H. and Newman, C.M., American Mathematical Society, 23–30.
  • [7] Jensen, S.T. and Rahbek, A. (2004a). Asymptotic inference for nonstationary GARCH., Econometric Theory 20, 1203–1226.
    Mathematical Reviews (MathSciNet): MR2101955
    Zentralblatt MATH: 1069.62067
    Digital Object Identifier: doi:10.1017/S0266466604206065
  • [8] Jensen, S. and Rahbek, A. (2004b). Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case., Econometrica 72, 641–646.
    Zentralblatt MATH: 1091.62074
    Digital Object Identifier: doi:10.1111/j.1468-0262.2004.00504.x
  • [9] Krengal, U. (1985)., Ergodic Theorems. W. de Gruyter, New York.
  • [10] Ledrappier, F. (1984). Quelques proprietes des exposants caracteristiques, in Ecole d’ete de probabilites de Saint-Flour XII - 1982, edited by Hennequin, P.L., Springer, Berlin, 305–396.
  • [11] LeJan, Y. (1991). A second order extension of Oseledets theorem, in Lyapunov exponents: proceedings of a conference held in Oberwolfach, May 28-June 2, 1990. Lecture Notes in Mathematics 1486, Arnold, L., Crauel, H. and Eckmann, J.P. (eds.), 81–85. Springer-Verlag, Berlin.
  • [12] Pollard, D. (1984)., Convergence of Stochastic Processes. Springer, New York.
    Zentralblatt MATH: 0544.60045
  • [13] Robinson, P.M. and Zaffaroni, P. (2006). Pseudo-maximum likelihood estimation of, ARCH(∞) models. Annals of Statistics 34, 1049–1074.
    Zentralblatt MATH: 1113.62107
    Digital Object Identifier: doi:10.1214/009053606000000245
    Project Euclid: euclid.aos/1152540742
  • [14] Rudin, W. (1987)., Real and Complex Analysis, 3rd Ed. McGraw-Hill, New York.
  • [15] Straumann, D. (2005)., Estimation in Conditionally Heteroscedastic Time Series Models. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2142271
    Zentralblatt MATH: 1086.62103

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