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RENORMING VOLATILITIES IN A FAMILY OF GARCH MODELS

  • Dong Li (a1) and Wuqing Wu (a2)
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.

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Corresponding author
*Address correspondence to Wuqing Wu, School of Business, Renmin University of China, Beijing 100872, China; e-mail: wwq@ruc.edu.cn.
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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).

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References
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Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed. Wiley.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327.
Bougerol, P. & Picard, N. (1992a) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115127.
Bougerol, P. & Picard, N. (1992b) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 17141730.
Brandt, A. (1986) The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients. Advances in Applied Probability 18, 211220.
Chan, N.H. & Ng, C.T. (2009) Statistical inference for non-stationary GARCH(p, q) models. Electronic Journal of Statistics 3, 956992.
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, 427431.
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, 83106.
Duan, J.C. (1997) Augmented GARCH(p, q) process and its diffusion limit. Journal of Econometrics 79, 97127.
Engle, R.F. & Bollerslev, T. (1986) Modelling the persistence of conditional variances. Econometric Reviews 5, 187.
Engle, R.F. & Ng, V.K. (1993) Measuring and testing the impact of news on volatility. Journal of Finance 48, 17491778.
Fornari, F. & Mele, A. (1997) Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 4965.
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, 815834.
Francq, C. & Zakoïan, J.-M. (2010) GARCH Models: Structure, Statistical Inference and Financial Applications. Wiley.
Francq, C. & Zakoïan, J.-M. (2012) Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models. Econometrica 80, 821861.
Francq, C. & Zakoïan, J.-M. (2013) Inference in nonstationary asymmetric GARCH models. Annals of Statistics 41, 19701998.
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, 17791801.
He, C. & Teräsvirta, T. (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173192.
Hitczenko, P. & Wesołowski, J. (2011) Renorming divergent perpetuities. Bernoulli 17, 880894.
Jensen, S.T. & Rahbek, A. (2004a) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641646.
Jensen, S.T. & Rahbek, A. (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 12031226.
Kristensen, D. (2009) On stationarity and ergodicity of the bilinear model with applications to GARCH models. Journal of Time Series Analysis 30, 125144.
Li, D., Li, M., & Wu, W. (2014) On dynamics of volatilities in nonstationary GARCH models. Statistics and Probability Letters 94, 8690.
Lin, Z.Y. & Lu, C.R. (1996) Limit Theory for Mixing Dependent Random Variables, Science Press and Kluwer Academic Publishers.
Ling, S. & McAleer, M. (2002a) Necessary and sufficient moment conditions for the GARCH(r, s) and asymmetric GARCH(r, s) models. Econometric Theory 18, 722729.
Ling, S. & McAleer, M. (2002b) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109117.
Linton, O., Pan, J., & Wang, H. (2010) Estimation for a nonstationary semi-strong GARCH(1,1) with heavy-tailed errors. Econometric Theory 26, 128.
Meitz, M. (2006) A necessary and sufficient condition for the strict stationarity of a family of GARCH processes. Econometric Theory 22, 985988.
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318334.
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 11291155.
Taylor, S. (2008) Modelling Financial Time Series, 2nd ed. World Scientific.
Yang, M. & Bewley, R. (1995) Moving average conditional heteroskedastic processes. Economics Letters 49, 367372.
Zakoïan, J.-M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931955.
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Econometric Theory
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