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Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. (English) Zbl 1108.62094

Summary: This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form \(X_t=\sigma_tZ_t\), where the unobservable volatility \(\sigma_t\) is a parametric function of \((X_{t-1},\dots,X_{t-p},\sigma_{t -1},\dots,\sigma_{t-q})\) for some \(p,q\geq 0\), and \((Z_t)\) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lip-schitz coefficient) property. These assumption are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution \((X_t)\) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
60H30 Applications of stochastic analysis (to PDEs, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60H25 Random operators and equations (aspects of stochastic analysis)
62F10 Point estimation
62M05 Markov processes: estimation; hidden Markov models
91B84 Economic time series analysis
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[1] Berkes, I. and Horváth, L. (2003). The rate of consistency of the quasi-maximum likelihood estimator. Statist. Probab. Lett. 61 133–143. · Zbl 1041.62017
[2] Berkes, I., Horváth, L. and Kokoszka, P. (2003). GARCH processes: Structure and estimation. Bernoulli 9 201–227. · Zbl 1064.62094
[3] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003
[4] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307–327. · Zbl 0865.62085
[5] Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM J. Control Optim. 31 942–959. · Zbl 0785.93040
[6] Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115–127. · Zbl 0746.62087
[7] Boussama, F. (1998). Ergodicité, mélange et estimation dans les modèles GARCH. Ph.D. dissertation, Univ. Paris VII.
[8] Boussama, F. (2000). Normalité asymptotique de l’estimateur du pseudo-maximum de vraisemblance d’un modèle GARCH. C. R. Acad. Sci. Paris Sér. I Math. 331 81–84. · Zbl 0952.62022
[9] Brandt, A. (1986). The stochastic equation \(Y_n+1=A_nY_n+B_n\) with stationary coefficients. Adv. in Appl. Probab. 18 211–220. JSTOR: · Zbl 0588.60056
[10] Breidt, F. J., Davis, R. A. and Trindade, A. (2001). Least absolute deviation estimation for all-pass time series models. Ann. Statist. 29 919–946. · Zbl 1012.62094
[11] Brockwell, P. and Davis, R. A. (1991). Time Series : Theory and Methods . Springer, New York. · Zbl 0709.62080
[12] Comte, F. and Lieberman, O. (2003). Asymptotic theory for multivariate GARCH processes. J. Multivariate Anal. 84 61–84. · Zbl 1038.62077
[13] Davis, R. A. and Dunsmuir, W. (1996). Maximum likelihood estimation for MA(1) processes with a root on or near the unit circle. Econometric Theory 12 1–29. JSTOR:
[14] Ding, Z., Granger, C. W. J. and Engle, R. (1993). A long memory property of stock market returns and a new model. J. Empirical Finance 1 83–106.
[15] Francq, C. and Zakoïan, J.-M. (2004). Maximum likelihood estimation of pure GARCH and ARMA–GARCH processes. Bernoulli 10 605–637. · Zbl 1067.62094
[16] Granger, C. W. J. and Andersen, A. (1978/79). On the invertibility of time series models. Stochastic Process. Appl. 8 87–92. · Zbl 0387.62076
[17] Hall, P. and Yao, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71 285–317. JSTOR: · Zbl 1136.62368
[18] Hannan, E. J. (1973). The asymptotic theory of linear time-series models. J. Appl. Probab. 10 130–145. [Correction 10 913.] JSTOR: · Zbl 0261.62073
[19] Jeantheau, T. (1998). Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14 70–86. JSTOR: · Zbl 04544644
[20] Krengel, U. (1985). Ergodic Theorems . de Gruyter, Berlin. · Zbl 0575.28009
[21] Lang, S. (1969). Analysis II . Addison–Wesley, Reading, MA. · Zbl 0176.00504
[22] Lee, S. and Hansen, B. (1994). Asymptotic theory for the GARCH\((1,1)\) quasi-maximum likelihood estimator. Econometric Theory 10 29–52. JSTOR:
[23] Lumsdaine, R. (1995). Finite-sample properties of the maximum likelihood estimator in GARCH\((1,1)\) and IGARCH\((1,1)\) models: A Monte Carlo investigation. J. Bus. Econom. Statist. 13 1–10. JSTOR:
[24] Lumsdaine, R. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH\((1,1)\) and covariance stationary GARCH\((1,1)\) models. Econometrica 64 575–596. JSTOR: · Zbl 0844.62080
[25] Mikosch, T. and Straumann, D. (2006). Stable limits of martingale transforms with application to the estimation of GARCH parameters. Ann. Statist. 34 493–522. · Zbl 1091.62082
[26] Nelson, D. (1990). Stationarity and persistence in the GARCH\((1,1)\) model. Econometric Theory 6 318–334. JSTOR:
[27] Nelson, D. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 347–370. JSTOR: · Zbl 0722.62069
[28] Pfanzagl, J. (1969). On the measurability and consistency of minimum contrast estimates. Metrika 14 249–272. · Zbl 0181.45501
[29] Ranga Rao, R. (1962). Relations between weak and uniform convergence of measures with applications. Ann. Math. Statist. 33 659–680. · Zbl 0117.28602
[30] Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models in Econometrics , Finance and Other Fields (D. R. Cox, D. Hinkley and O. E. Barndorff-Nielsen, eds.) 1–67. Chapman and Hall, London.
[31] Straumann, D. (2005). Estimation in Conditionally Heteroscedastic Time Series Models . Lecture Notes in Statist. 181 . Springer, Berlin. · Zbl 1086.62103
[32] Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595–601. · Zbl 0034.22902
[33] Zakoïan, J.-M. (1994). Threshold heteroscedastic models. J. Econom. Dynam. Control 18 931–955. · Zbl 0875.90197
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