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.
  • [2] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [3] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM J. Control Optim. 31 942–959.
  • [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.
  • [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.
  • [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.
  • [9] Doukhan, P. and Wintenberger, O. (2008). Weakly dependent chains with infinite memory. Stochastic Process. Appl. 118 1997–2013.
  • [10] Duflo, M. (1997). Random Iterative Models. Applications of Mathematics (New York) 34. Springer, Berlin.
  • [11] Engle, R. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987–1007.
  • [12] Engle, R. and Kroner, K. (1995). Multivariate simultaneous generalized ARCH. Econometric Theory 11 122–150.
  • [13] Feller, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
  • [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.
  • [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.
  • [18] Jeantheau, T. (2004). A link between complete models with stochastic volatility and arch models. Finance Stoch. 8 111–132.
  • [19] Krengel, U. (1985). Ergodic Theorems 6. De Gruyter, Berlin.
  • [20] Ling, S. and McAleer, M. (2003). Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19 280–310.
  • [21] Pfanzagl, J. (1969). On the mesurability and consistency of minimum contrast estimates. Metrika 14 249–334.
  • [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.
  • [24] Robinson, P. and Zaffaroni, P. (2006). Pseudo-maximum likelihood estimation of ARCH(∞) models. Ann. Statist. 34 1049–1074.
  • [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.