The Annals of Statistics

Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes

Jean-Marc Bardet and Olivier Wintenberger

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Abstract

Strong consistency and asymptotic normality of the quasi-maximum likelihood estimator are given for a general class of multidimensional causal processes. For particular cases already studied in the literature [for instance univariate or multivariate ARCH(∞) processes], the assumptions required for establishing these results are often weaker than existing conditions. The QMLE asymptotic behavior is also given for numerous new examples of univariate or multivariate processes (for instance TARCH or NLARCH processes).

Article information

Source
Ann. Statist., Volume 37, Number 5B (2009), 2730-2759.

Dates
First available in Project Euclid: 17 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1247836667

Digital Object Identifier
doi:10.1214/08-AOS674

Mathematical Reviews number (MathSciNet)
MR2541445

Zentralblatt MATH identifier
1173.62063

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F12: Asymptotic properties of estimators

Keywords
Quasi-maximum likelihood estimator strong consistency asymptotic normality multidimensional causal processes multivariate ARMA–GARCH processes

Citation

Bardet, Jean-Marc; Wintenberger, Olivier. Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. Ann. Statist. 37 (2009), no. 5B, 2730--2759. doi:10.1214/08-AOS674. https://projecteuclid.org/euclid.aos/1247836667


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References

  • [1] Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9 201–227.
    Zentralblatt MATH: 1064.62094
    Digital Object Identifier: doi:10.3150/bj/1068128975
    Project Euclid: euclid.bj/1068128975
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR233396
    Zentralblatt MATH: 0172.21201
  • [3] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM J. Control Optim. 31 942–959.
    Mathematical Reviews (MathSciNet): MR1227540
    Zentralblatt MATH: 0785.93040
    Digital Object Identifier: doi:10.1137/0331041
  • [4] Boussama, F. (1998). Ergodicity, mixing and estimation in GARCH models. Ph.D. thesis, Univ. de Paris 07, Paris.
  • [5] Comte, F. and Lieberman, O. (2003). Asymptotic theory for multivariate GARCH processes. J. Multivariate Anal. 84 61–84.
    Mathematical Reviews (MathSciNet): MR1965823
    Zentralblatt MATH: 1038.62077
    Digital Object Identifier: doi:10.1016/S0047-259X(02)00009-X
  • [6] Ding, Z., Granger, C. W. and Engle, R. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1 83–106.
  • [7] Douc, R., Roueff, F. and Soulier, P. (2008). On the existence of some ARCH(∞) processes. Stochastic Process. Appl. 118 755–761.
    Mathematical Reviews (MathSciNet): MR2411519
    Zentralblatt MATH: 1136.60327
    Digital Object Identifier: doi:10.1016/j.spa.2007.06.002
  • [8] Doukhan, P., Teyssière, G. and Winant, P. (2006). A LARCH(∞) vector valued process. In Dependence in Probability and Statistics (P. Bertail, P. Doukhan and P. Soulier, eds.) 245–258. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2283258
    Zentralblatt MATH: 1113.60038
    Digital Object Identifier: doi:10.1007/0-387-36062-X_11
  • [9] Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory. Stochastic Process. Appl. 118 1997–2013.
    Mathematical Reviews (MathSciNet): MR2462284
    Zentralblatt MATH: 1166.60031
    Digital Object Identifier: doi:10.1016/j.spa.2007.12.004
  • [10] Duflo, M. (1997). Random Iterative Models. Applications of Mathematics (New York) 34. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1485774
    Zentralblatt MATH: 0868.62069
  • [11] Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007.
    Mathematical Reviews (MathSciNet): MR666121
    Digital Object Identifier: doi:10.2307/1912773
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0491.62099
  • [12] Engle, R. and Kroner, K. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory 11 122–150.
    Mathematical Reviews (MathSciNet): MR1325104
    Digital Object Identifier: doi:10.1017/S0266466600009063
    JSTOR: links.jstor.org
  • [13] Feller, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR210154
    Zentralblatt MATH: 0138.10207
  • [14] Francq, C. and Zakoïan, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10 605–637.
    Zentralblatt MATH: 1067.62094
    Digital Object Identifier: doi:10.3150/bj/1093265632
    Project Euclid: euclid.bj/1093265632
  • [15] Giraitis, L., Kokoszka, P. and Leipus, R. (2000). Stationary arch models: Dependence structure and central limit theorem. Econometric Theory 16 3–22.
    Mathematical Reviews (MathSciNet): MR1749017
    Zentralblatt MATH: 0986.60030
    Digital Object Identifier: doi:10.1017/S0266466600161018
    JSTOR: links.jstor.org
  • [16] Jeantheau, T. (1993). Modèles autorégressifs à erreur conditionnellement hétéroscédastique. Ph.D. thesis, Univ. Paris VII.
  • [17] Jeantheau, T. (1998). Strong consistency of estimators for multivariate arch models. Econometric Theory 14 70–86.
    Mathematical Reviews (MathSciNet): MR1613694
    Digital Object Identifier: doi:10.1017/S0266466698141038
    JSTOR: links.jstor.org
  • [18] Jeantheau, T. (2004). A link between complete models with stochastic volatility and arch models. Finance Stoch. 8 111–132.
    Mathematical Reviews (MathSciNet): MR2022981
    Zentralblatt MATH: 1098.91052
    Digital Object Identifier: doi:10.1007/s00780-003-0103-6
  • [19] Krengel, U. (1985). Ergodic Theorems 6. De Gruyter, Berlin.
    Mathematical Reviews (MathSciNet): MR797411
    Zentralblatt MATH: 0575.28009
  • [20] Ling, S. and McAleer, M. (2003). Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19 280–310.
    Mathematical Reviews (MathSciNet): MR1966031
    JSTOR: links.jstor.org
    Zentralblatt MATH: 06993481
  • [21] Pfanzagl, J. (1969). On the mesurability and consistency of minimum contrast estimates. Metrika 14 249–334.
    Mathematical Reviews (MathSciNet): MR365797
    Digital Object Identifier: doi:10.1007/BF01893900
    Zentralblatt MATH: 0181.45501
  • [22] Rabemananjara, R. and Zakoïan, J. (1993). Threshold ARCH models and asymmetries in volatility. J. Appl. Econometrics 8 31–49.
  • [23] Robinson, P. (1991). Testing for strong serial correlation and dynamic conditional heteroscedasticity in multiple regression. J. Econometrics 47 67–84.
    Mathematical Reviews (MathSciNet): MR1087207
    Digital Object Identifier: doi:10.1016/0304-4076(91)90078-R
    Zentralblatt MATH: 0734.62070
  • [24] Robinson, P. and Zaffaroni, P. (2006). Pseudo-maximum likelihood estimation of ARCH(∞) models. Ann. Statist. 34 1049–1074.
    Mathematical Reviews (MathSciNet): MR2278351
    Zentralblatt MATH: 1113.62107
    Digital Object Identifier: doi:10.1214/009053606000000245
    Project Euclid: euclid.aos/1152540742
  • [25] Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Ann. Statist. 34 2449–2495.
    Mathematical Reviews (MathSciNet): MR2291507
    Zentralblatt MATH: 1108.62094
    Digital Object Identifier: doi:10.1214/009053606000000803
    Project Euclid: euclid.aos/1169571804

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