Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-19T23:17:13.903Z Has data issue: false hasContentIssue false
  • English
  • Français

RENORMING VOLATILITIES IN A FAMILY OF GARCH MODELS

Published online by Cambridge University Press:  11 December 2017

Dong Li  and
Wuqing Wu
Dong Li
Affiliation:
Tsinghua University
Wuqing Wu*
Affiliation:
Renmin University of China
*
*Address correspondence to Wuqing Wu, School of Business, Renmin University of China, Beijing 100872, China; e-mail: wwq@ruc.edu.cn.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the weak convergence of renorming volatilities in a family of GARCH(1,1) models from a functional point of view. After suitable renormalization, it is shown that the limiting distribution is a geometric Brownian motion when the associated top Lyapunov exponent γ > 0 and is an exponential functional of the maximum process of a Brownian motion when γ = 0. This indicates that the volatility of the GARCH(1,1)-type model has a completely different random structure according to the sign of γ. The obtained results further strengthen our understanding of volatilities in GARCH-type models. Simulation studies are carried out to assess our findings.

Type
MISCELLANEA
Information
Econometric Theory , Volume 34 , Issue 6 , December 2018 , pp. 1370 - 1382
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We would like to thank the co-editor Dennis Kristensen and two anonymous referees whose comments greatly improved the paper. We also thank the Editor Peter C.B. Phillips for pointing out many syntax errors in an earlier version. Li’s research is partially supported by the Start-up Fund of Tsinghua University (No. 53330230117) and the NSFC (No. 11401337, No. 11571348 and No. 11771239). Wu’s research is supported by the Research Funds of Renmin University of China (No.16XNB025), the Social Science Foundation of Beijing (No. 17GLB022) and the NSFC (No. 71631004 (Key Project) and No. 71403251).

References

REFERENCES

Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed. Wiley.CrossRefGoogle Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992a) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Bougerol, P. & Picard, N. (1992b) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1730.Google Scholar
Brandt, A. (1986) The stochastic equation Y n+1 = A nY n + B n with stationary coefficients. Advances in Applied Probability 18, 211–220.Google Scholar
Chan, N.H. & Ng, C.T. (2009) Statistical inference for non-stationary GARCH(p, q) models. Electronic Journal of Statistics 3, 956–992.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Ding, Z., Granger, C.W.J., & Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.CrossRefGoogle Scholar
Duan, J.C. (1997) Augmented GARCH(p, q) process and its diffusion limit. Journal of Econometrics 79, 97–127.CrossRefGoogle Scholar
Engle, R.F. & Bollerslev, T. (1986) Modelling the persistence of conditional variances. Econometric Reviews 5, 1–87.CrossRefGoogle Scholar
Engle, R.F. & Ng, V.K. (1993) Measuring and testing the impact of news on volatility. Journal of Finance 48, 1749–1778.CrossRefGoogle Scholar
Fornari, F. & Mele, A. (1997) Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 49–65.3.0.CO;2-6>CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2012) Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models. Econometrica 80, 821–861.Google Scholar
Francq, C. & Zakoïan, J.-M. (2013) Inference in nonstationary asymmetric GARCH models. Annals of Statistics 41, 1970–1998.CrossRefGoogle Scholar
Glosten, L.R., Jaganathan, R., & Runkle, D. (1993) On the relation between the expected values and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
He, C. & Teräsvirta, T. (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.CrossRefGoogle Scholar
Hitczenko, P. & Wesołowski, J. (2011) Renorming divergent perpetuities. Bernoulli 17, 880–894.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004a) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.Google Scholar
Jensen, S.T. & Rahbek, A. (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.CrossRefGoogle Scholar
Kristensen, D. (2009) On stationarity and ergodicity of the bilinear model with applications to GARCH models. Journal of Time Series Analysis 30, 125–144.CrossRefGoogle Scholar
Li, D., Li, M., & Wu, W. (2014) On dynamics of volatilities in nonstationary GARCH models. Statistics and Probability Letters 94, 86–90.CrossRefGoogle Scholar
Lin, Z.Y. & Lu, C.R. (1996) Limit Theory for Mixing Dependent Random Variables, Science Press and Kluwer Academic Publishers.Google Scholar
Ling, S. & McAleer, M. (2002a) Necessary and sufficient moment conditions for the GARCH(r, s) and asymmetric GARCH(r, s) models. Econometric Theory 18, 722–729.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2002b) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109–117.CrossRefGoogle Scholar
Linton, O., Pan, J., & Wang, H. (2010) Estimation for a nonstationary semi-strong GARCH(1,1) with heavy-tailed errors. Econometric Theory 26, 1–28.CrossRefGoogle Scholar
Meitz, M. (2006) A necessary and sufficient condition for the strict stationarity of a family of GARCH processes. Econometric Theory 22, 985–988.CrossRefGoogle Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 1129–1155.CrossRefGoogle Scholar
Taylor, S. (2008) Modelling Financial Time Series, 2nd ed. World Scientific.Google Scholar
Yang, M. & Bewley, R. (1995) Moving average conditional heteroskedastic processes. Economics Letters 49, 367–372.CrossRefGoogle Scholar
Zakoïan, J.-M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar