Strict stationarity testing and estimation of explosive ARCH models

Francq, Christian and Zakoian, Jean-Michel (2010): Strict stationarity testing and estimation of explosive ARCH models.

Abstract

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of ARCH(1) models without strict stationarity constraints, and considers applications to testing problems. The estimator is unrestricted, in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the ARCH coefficient is obtained at the boundary of the stationarity region, but this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and non stationary situations, but is consistently estimated, with the same estimator, in both cases. Tests of strict stationarity and non stationarity are proposed. Their behaviors are studied under the null assumption and under local alternatives. The tests developed for the ARCH(1) model are able to detect non-stationarity in more general GARCH models. A numerical illustration based on stock indices is proposed.

• Item Type: MPRA Paper
• Language: English
• Keywords: ARCH model; Inconsistency of estimators; Local power of tests; Nonstationarity; Quasi Maximum Likelihood Estimation
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Aue A., and L. Horv\'ath (2009) Quasi-likelihood Estimation in Stationary and Nonstationary Autoregressive Models With Random Coefficients. Preprint

Berkes, I., Horv\'ath, L. and P.S. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201--227.

Billingsley P. (1995) Probability and Measure. John Wiley, New York.

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307--327.

Drost, F. C. and C. A. J. Klaassen (1997) Efficient estimation in semiparametric GARCH models. Journal of Econometrics 81, 193--221.

Drost, F.C., Klaassen, C.A.J. and B.J.M. Werker (1997) Adaptive estimation in time-series models. Annals of Statistics 25, 786--817.

Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 987--1007.

Francq, C. and J-M. Zakoïan (2004) Maximum Likelihood Estimation of Pure GARCH and ARMA-GARCH Processes. Bernoulli 10, 605--637.

Francq, C. and J-M. Zakoïan (2006a) Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process. Econometric Theory 22, 815--834.

Francq, C. and J-M. Zakoïan (2006b) On efficient inference in GARCH processes. In: Bertail P, Doukhan P, Soulier P (eds) Statistics for dependent data. Springer, New-York: 305--327

Goldie, C.M. and R.A. Maller (2000) Stability of perpetuities. Annals of Probability 28, 1195--1218.

Hörmann, S. (2008) Augmented GARCH sequences: dependence structure and asymptotics. Bernoulli 14, 543--561.

Jensen, S.T. and A. Rahbek (2004a) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641--646.

Jensen, S.T. and A. Rahbek (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203--1226.

Klüppelberg, C., Lindner, A. and R. Maller (2004) A continuous time GARCH Process driven by a Lévy process: stationarity and second order behaviour. Journal of Applied Probability 41, 601--622.

Lee, S. and M. Taniguchi (2005) Asymptotic theory for ARCH-SM models: LAN and residual empirical processes. Statistica Sinica 15, 215--234.

Ling, S. and M. McAleer (2003) Adaptative estimation in nonstationary ARMA models with GARCH errors. The Annals of Statistics 31, 642--674.

Linton, O., Pan, J. and H. Wang (2009) Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. To appear in Econometric Theory.

Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318--334.

van der Vaart, A.W. (1998) Asymptotic statistics. Cambridge University Press, United Kingdom.