Wintenberger, Olivier and Cai, Sixiang (2011): Parametric inference and forecasting in continuously invertible volatility models.

PDF
MPRA_paper_31767.pdf Download (951kB)  Preview 
Abstract
We introduce the notion of continuously invertible volatility models that relies on some Lyapunov condition and some regularity condition. We show that it is almost equivalent to the volatilities forecasting efﬁciency of the parametric inference approach based on the Stochastic Recurrence Equation (SRE) given in Straumann (2005). Under very weak assumptions, we prove the strong consistency and the asymptotic normality of an estimator based on the SRE. From this parametric estimation, we deduce a natural forecast of the volatility that is strongly consistent. We successfully apply this approach to recover known results on univariate and multivariate GARCH type models where our estimator coincides with the QMLE. In the EGARCH(1,1)model, we apply this approach to ﬁnd a strongly consistence forecast and to prove that our estimator is asymptotically normal when the limiting covariance matrix exists. Finally, we give some encouraging empirical results of our approach on simulations and real data.
Item Type:  MPRA Paper 

Original Title:  Parametric inference and forecasting in continuously invertible volatility models 
Language:  English 
Keywords:  Invertibility, volatility models, parametric estimation, strong consistency, asymptotic normality, asymmetric GARCH, exponential GARCH, stochastic recurrence equation, stationarity 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C13  Estimation: General C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C53  Forecasting and Prediction Methods ; Simulation Methods C  Mathematical and Quantitative Methods > C0  General > C01  Econometrics 
Item ID:  31767 
Depositing User:  Sixiang CAI 
Date Deposited:  22 Jun 2011 12:36 
Last Modified:  18 Sep 2016 12:05 
References:  BARDET, J. M. and WINTENBERGER, O. (2009). Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes. Ann. Statist. 37 27302759. BERKES, I., HORVATH, L. and KOKOSZKA, P. (2003). GARCH processes: struc ture and estimation. Bernoulli 9 201227. BOLLERSLEV, T. (1986). Generalized autoregressive conditional heteroskedas ticity. Journal of Econometrics 31 307327. BOLLERSLEV, T. P. (1990). Modeling the coherence in short run nominal ex change rates: A multivariate generalized ARCH approach. Review of Eco nomics and Statistics 72 498505. BOUCHAUD, J.P. and POTTERS, M. (2001).More stylized facts of ﬁnancialmar kets: leverage effect and downside correlations. Physica A: StatisticalMechan ics and its Applications 299 60  70. BOUGEROL, P. (1993). Kalman ﬁltering with random coefﬁcients and contrac tions. SIAM J. Control and Optimization 31 942959. BOUGEROL, P. and PICARD, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115–127. BRANDT, M. W. and JONES, C. S. (2006). Volatility forecasting with range based EGARCH Models. Journal of Business & Economic Statistics 24 470486. CONT, R. (2001). Empirical properties of asset returns: stylized facts and statis tical issues. Quantitative Finance 1 223–236. DEMOS, A. and KYRIAKOPOULOU, D. (2009). Asymptotic expansions of the QMLEs in the EGARCH(1,1) Model. preprint. 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 83106. ENGLE, R. F. (1982). Autoregressive conditional heteroscedasticity with esti mates of the variance of united kingdom inﬂation. Econometrica 50 9871007. FRANCQ, C. and ZAKOÏAN, J. M. (2004). Maximum likelihood estimation of pure GARCH and ARMAGARCH processes. Bernoulli 10 605637. FRANCQ, C. and ZAKOÏAN, J. M. (2011). QML estimation of a class of multi variate asymmetric GARCH models. forthcoming in Econometric Theory. GLOSTEN, L. R., JAGANNATHAN, R. and RUNKLE, D. E. (1993).On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48 17791801. HARVEY, A. (2010). Exponential conditional volatility models. Cambridge Working Papers in Economics report No. 1040, Faculty of Economics, Uni versity of Cambridge. HE, C., TERÄSVIRTA, T. and MALMSTEN, H. (2002). Moment structure of a family of ﬁrstorder exponential GARCH models. Econometric Theory 18 868 885. JEANTHEAU, T. (1993). Modèles autorégressifs à erreur conditionellement hétéroscédastique. PhD thesis, Université Paris VII. JEANTHEAU, T. (1998). Strong consistency of estimation for multivariate ARCH models. Econometric Theory 14 7086. NELSON, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6 318334. NELSON, D. B. (1991). Conditional Heteroskedasticity in Asset Returns : A New Approach. Econometrica 59 347370. PATTON, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. J. Econometrics 160 246  256. RODRIGUEZ, M. J. and RUIZ, E. (2009). GARCH models with leverage effect: differences and similarities. Statistics and Econometrics Working Papers re port No. ws090302, Universidad Carlos III, Departamento de Estadestica y Econometre. SOROKIN, A. (2011). Noninvertibility in some heteroscedastic models. Arxiv preprint #1104.3318. STRAUMANN, D. (2005). Estimation in Conditionally Heteroscedastic Time Series Models. Lectures Notes in Statistics 181. Springer, New York. STRAUMANN, D. and MIKOSCH, T. (2006). Quasimaximumlikelihood esti mation in conditionally heteroscedastic time series: a stochastic recurrence equation approach. Ann. Statist. 34 24492495. TONG, H. (1993). NonLinear Time Series, A Dynamical System Approach. Oxford Statistical Science Series 6. Oxford Science Publications, Oxford. ZAFFARONI, P. (2009). Whittle estimation of EGARCH and other exponential volatility models. J. Econometrics 151 190200. ZAKOÏAN, J. M. (1994). Threshold heteroscedastic models. J. Econom. Dynam. Control 18 931955. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/31767 