Abstract
Let Yt be an autoregressive process with order one, i.e., Yt = μ + ϕnYt−1 + εt, where [εt] is a heavy tailed general GARCH noise with tail index α. Let \({{\hat \phi }_n}\) be the least squares estimator (LSE) of ϕn For μ = 0 and α < 2, it is shown by Zhang and Ling (2015) that \({{\hat \phi }_n}\) is inconsistent when Yt is stationary (i.e., ϕn ≡ ϕ < 1), however, Chan and Zhang (2010) showed that \({{\hat \phi }_n}\) is still consistent with convergence rate n when Yt is a unit-root process (i.e., ϕn = 1) and [εt] is a GARCH(1, 1) noise. There is a gap between the stationary and nonstationary cases. In this paper, two important issues will be considered: (1) what about the nearly unit root case? (2) When can ϕ be estimated consistently by the LSE? We show that when ϕn = 1 − c/n, then \({{\hat \phi }_n}\) converges to a functional of stable process with convergence rate n. Further, we show that if limn→∞kn(1 − ϕn) = c for a positive constant c, then \({k_n}({\hat \phi _n} - \phi )\) converges to a functional of two stable variables with tail index α/2, which means that ϕn can be estimated consistently only when kn → ∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B Basrak, R A Davis, T Mikosch. Regular variation of GARCH processes, Stoch Proc Appl, 2002, 99(1): 95–115.
T Bollerslev. Generalized autoregressive conditional heteroskedasticity, J Econometrics, 1986, 31(3): 307–327.
N H Chan. Time series with roots on or near the unit circle, In: Mikosch T, Kreiß JP, Davis R, Andersen T, (eds), Handbook of Financial Time Series, Springer-Verlag, New York, 2009.
N H Chan, L Peng. Weighted least absolute deviations estimation for AR(1) process with ARCH(1) errors, Biometrika, 2005, 92(2): 477–484.
N H Chan, R M Zhang. Inference for nearly nonstationary processes under strong dependence with infinite variance, Stat Sinica, 2009, 19: 925–947.
N H Chan, R M Zhang. Inference for unit-root models with infinite variance GARCH errors, Stat Sinica, 2010, 20: 1363–1393.
R F Engle. Discussion: stock market volatility and the crash of 1987, Rev Financ Stud, 1990, 3(1): 103–106.
F Fornari, A Mele. Sign- and volatility-switching ARCH models: theory and applications to international stock markets, J Appl Econom, 1997, 12: 49–65.
C Francq, J M Zakoïan. Mixing properties of a general class of GARCH(1, 1) models without moment assumptions on the observed process, Econometric Theory, 2006, 22(5): 815–834.
P Hall, Q W Yao. Inference in ARCH and GARCH models with heavy-tailed errors, Econometrica, 2003, 71(1): 285–317.
C He, T Terasvirta. Properties of moments of a family of GARCH processes, J Econometrics, 1999, 92(1): 173–192.
H Kesten. Random difference equations and renewal theory for products of random matrices, Acta Math, 1973, 131: 207–248.
K Knight. Limit theory for autoregressive-parameter estimates in an infinite-variance random walk, Can J Stat, 1989, 17(3): 261–278.
G D Li, W K Li. Least Absolute Deviations Estimation for unit root processes with GARCH errors, Ecomomet Theor, 2009, 25(5): 1208–1227.
S Ling, W K Li. Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors, Ann Stat, 1998, 26(1): 84–125.
S Ling, W K Li. Asymptotic inference for unit-root with GARCH (1,1) errors, Economet Theor, 2003, 19(4): 541–564.
S Ling, W K Li. Estimation and testing for a unit-root process with GARCH (1,1) errors, Economet Rev, 2003, 22(2): 179–202.
D B Nelson. Stationary and persistence in GARCH (1, 1) models, Economet Theor, 1990, 6(3): 318–334.
L Peng, Q W Yao. Least absolute deviations estimation for ARC and GARCH models, Biometrika, 2003, 90(4): 967–975.
P C B Phillips. A shortcut to LAD asymptotics, Economet Theor, 1991, 7: 450–463.
P C B Phillips, T Magdalinos. Limit theory for moderate deviations from a unit root, J Econometrics, 2007, 136(1): 115–130.
G W Schwert. Why does stock market volatility change over time? J Financ, 1989, 44(5): 1115–1153.
E Sentana. Quadratic ARCH models, Rev Econ Stud, 1995, 62(4): 639–661.
S Taylor. Modelling Financial Time Series, Wiley, New York, 1986.
J M Zakoian. Threshold heteroskedastic models, J Econ Dyn Control, 1994, 18(5): 931–955.
R M Zhang, S Ling. Asymptotic inference for AR models with heavy-tailed G-GARCH noises, Economet Theor, 2015, 31(4): 880–890.
R M Zhang, C Y Sin, S Ling. On functional limits of short- and long-memory linear processes with GARCH(1,1) noises, Stoch Proc Appl, 2015, 125(2): 482–512.
R M Zhang, N H Chan. Nonstationary Linear Processes with Infinite Variance GARCH Errors, Economet Theor, 2021, 37(5): 892–925.
K Zhu, S Ling. LADE-based inference for ARMA models with unspecified and heavy-tailed heteroscedastic noises, J Am Stat Assoc, 2015, 110(510): 784–794.
Acknowledgement
We would like to thank the Editor and the anonymous referees for their critical comments and thoughtful suggestions, which led to a much improved version of this paper.
Funding
Zhang is supported by the National Natural Science Foundation of China(11771390, 12171427), ZPNS-FC(LZ21A010002), Fundamental Research Funds for the Central Universities (2021XZZX002) and Shi is supported by Natural Science Foundation of Fujian Province(2020J01794), Fujian Key Laboratory of Granular Computing and Applications(Minnan Normal University).
Author information
Authors and Affiliations
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zhang, Rm., Liu, Qm. & Shi, Jh. Nearly nonstationary processes under infinite variance GARCH noises. Appl. Math. J. Chin. Univ. 37, 246–257 (2022). https://doi.org/10.1007/s11766-022-4442-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-022-4442-5