Published online by Cambridge University Press: 11 December 2017
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.
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).