Inference in non stationary asymmetric garch models

Francq, Christian and Zakoian, Jean-Michel (2013): Inference in non stationary asymmetric garch models.

Abstract

This paper considers the statistical inference of the class of asymmetric power-transformed GARCH(1,1) models in presence of possible explosiveness. We study the explosive behavior of volatility when the strict stationarity condition is not met. This allows us to establish the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the parameter, including the power but without the intercept, when strict stationarity does not hold. Two important issues can be tested in this framework: asymmetry and stationarity. The tests exploit the existence of a universal estimator of the asymptotic covariance matrix of the QMLE. By establishing the local asymptotic normality (LAN) property in this nonstationary framework, we can also study optimality issues.

• Item Type: MPRA Paper
• Language: English
• Keywords: GARCH models; Inconsistency of estimators; Local power of tests; Nonstationarity; Quasi Maximum Likelihood Estimation
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: 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

References

Aknouche, A., Al-Eid, E.M. and A.M. Hmeid (2011) Offline and online weighted least squares estimation of nonstationary power ARCH processes. Statistics and Probability Letters 81, 1535--1540.

Aknouche, A. and E.M. Al-Eid (2012) Asymptotic inference of unstable periodic ARCH processes. Statistical Inference for Stochastic Processes 15, 61--79.

Aue A., and L. Horv\'ath (2011) Quasi-likelihood Estimation in Stationary and Nonstationary Autoregressive Models With Random Coefficients. Statistica Sinica, 21, 973--999.

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.

Bougerol, P. and Picard, N. (1992) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714--1729.

Dickey, D. A. and W. A. Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427--431.

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

Francq, C. and J-M. Zakoïan (2010) GARCH models: structure, statistical inference and financial applications. Chichester: John Wiley.

Francq, C. and J.M. Zakoïan (2012) Strict stationarity testing and estimation of explosive and stationary GARCH models. Econometrica 80, 821--861.

Glosten, L.R., Jaganathan, R. and D. Runkle (1993) On the relation between the expected values and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779--1801.

Hamadeh, T., and J.M. Zakoïan (2011) Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH Processes. Journal of Statistical Planning and Inference, 141, 488--507.

Higgins, M.L. and A.K. Bera (1992) A class of nonlinear ARCH models. International Economic Review, 33, 137-158.

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

Hwang, S.Y. and T.Y. Kim (2004) Power transformation and threshold modeling for ARCH innovations with applications to test for ARCH structure. Stochastic Processes and Their Applications 110, 295--314.

Karlsen H.A., Myklebust, T. and D. Tjøstheim (2007) Nonparametric estimation in a nonlinear cointegration type model. Annals of Statistics 35, 252--299.

Karlsen H.A. and D. Tjøstheim (2001) Nonparametric estimation in null recurrent time series. Annals of Statistics 29, 372--416.

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.

Li, C.W. and W.K. Li (1996) On a double-threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics 11, 253--274.

Ling, S. and D. Li (2008) Asymptotic inference for a nonstationary double AR(1) model. Biometrika 95, 257--263.

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

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

Pan, J., Wang, H., and H. Tong (2008) Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics, 142, 352-378.

Phillips, P.C.B. and P. Perron (1988) Testing for a Unit Root in Time Series Regression Biometrika, 75, 335--346.

Rabemananjara, R. and J-M. Zakoïan (1993) Threshold ARCH Models and Asymmetries in Volatility. Journal of Applied Econometrics 8, 31--49.

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