Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes
Jean-Marc Bardet and Olivier Wintenberger
Source: Ann. Statist. Volume 37, Number 5B
(2009), 2730-2759.
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).
First Page:
Show
Hide
Keywords: Quasi-maximum likelihood estimator; strong consistency; asymptotic normality; multidimensional causal processes; multivariate ARMA–GARCH processes
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1247836667
Digital Object Identifier: doi:10.1214/08-AOS674
Zentralblatt MATH identifier: 05596919
Mathematical Reviews number (MathSciNet): MR2541445
References
[1] Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9 201–227.
[2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Mathematical Reviews (MathSciNet): MR233396
[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
[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
[14] Francq, C. and Zakoïan, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10 605–637.
[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
[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
[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
[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
[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
