Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-25T06:03:11.763Z Has data issue: false hasContentIssue false
  • English
  • Français

NONSTATIONARY LINEAR PROCESSES WITH INFINITE VARIANCE GARCH ERRORS

Published online by Cambridge University Press:  26 October 2020

Rongmao Zhang  and
Ngai Hang Chan
Rongmao Zhang
Affiliation:
Zhejiang University
Ngai Hang Chan*
Affiliation:
Southwestern University of Finance The Chinese University of Hong Kong
*
Address correspondence to Ngai Hang Chan, Department of Statistics, The Chinese University of Hong Kong, Hong Kong; e-mail: nhchan@sta.cuhk.edu.hk.
Get access Rights & Permissions [Opens in a new window]

Abstract

Recently, Cavaliere, Georgiev, and Taylor (2018, Econometric Theory 34, 302–348) (CGT) considered the augmented Dickey–Fuller (ADF) test for a unit-root model with linear noise driven by i.i.d. infinite variance innovations and showed that ordinary least squares (OLS)-based ADF statistics have the same distribution as in Chan and Tran (1989, Econometric Theory 5, 354–362) for i.i.d. infinite variance noise. They also proposed an interesting question to extend their results to the case with infinite variance GARCH innovations as considered in Zhang, Sin, and Ling (2015, Stochastic Processes and their Applications 125, 482–512). This paper addresses this question. In particular, the limit distributions of the ADF for random walk models with short-memory linear noise driven by infinite variance GARCH innovations are studied. We show that when the tail index $\alpha <2$ , the limit distributions are completely different from that of CGT and the estimator of the parameters of the lag terms used in the ADF regression is not consistent. This paper provides a broad treatment of unit-root models with linear GARCH noises, which encompasses the commonly entertained unit-root IGARCH model as a special case.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 5 , October 2021 , pp. 892 - 925
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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 Editor, Professor Peter C.B. Phillips and the Co-Editor, Professor G. Cavaliere, and two anonymous referees for their critical comments and thoughtful suggestions, which led to a much improved version of this paper. Research supported in part by grants from NSFC (No. 11771390/11371318), the USyd-ZJU Partnership Collaboration Awards and the Fundamental Research Funds for the Central Universities (Zhang) and HKSAR-RGC-GRF Nos 14308218 and 14325216 and HKSR-RGC-TRF No. T32-101/15-R (Chan).

References

REFERENCES

Andersen, T.G., Davis, R.A., Kreiss, J., & Mikosch, T. (2009) Springer Handbook of Financial Time Series. Springer.Google Scholar
Avram, F. & Taqqu, M.S. (1992) Weak convergence of sums of moving averages in the $\alpha$ -stable domain of attraction. Annals of Probability 20, 483–503.CrossRefGoogle Scholar
Basrak, B., Davis, R.A., & Mikosch, T. (2002) Regular variation of GARCH processes. Stochastic Processes and their Applications 99, 95–115.CrossRefGoogle Scholar
Bayraktar, E., Horst, U., & Sircar, K.R. (2006) A limiting theorem for financial markets with inert investors. Mathematics of Operations Research 31, 789–810.CrossRefGoogle Scholar
Brillinger, D. (1981) Time Series: Data Analysis and Theory. Holden Day.Google Scholar
Cavaliere, G., Georgiev, I., & Taylor, A.M.R. (2018) Unit root inference for non-stationary linear processes driven by infinite variance innovations. Econometric Theory 34, 302–348.CrossRefGoogle Scholar
Chan, N.H. (2008) Time series with roots on or near the unit circle. In Andersen, T.G., Davis, R.A., Kreiss, J. and Mikosch, T. (eds.), Springer Handbook of Financial Time Series, Springer.Google Scholar
Chan, N.H., Peng, L., & Qi, Y.C. (2006) Quantile inference for near-integrated autoregressive time series with infinite variance. Statistica Sinica 16, 15–28.Google Scholar
Chan, N.H. & Tran, L.T. (1989) On the first order autoregressive process with infinite variance. Econometric Theory 5, 354–362.CrossRefGoogle Scholar
Chan, N.H. & Zhan, R.M. (2010) Inference for unit-root models with infinite variance GARCH errors. Statistica Sinica 20, 1363–1393.Google Scholar
Chan, N.H. & Zhang, R.M. (2009) Inference for nearly nonstationary processes under strong dependence with infinite variance. Statistica Sinica 19, 925–947.Google Scholar
Davis, R.A. & Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.CrossRefGoogle Scholar
Finkenstädt, B. & Rootzén, H. (2004) Extreme Values in Finance, Telecommunications, and the Environment. Chapman and Hall/CRC Press.Google 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
Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. Annals of Applied Probability 1, 126–166.CrossRefGoogle Scholar
Hasan, M.N. (2001) Rank tests of unit root hypothesis with infinite variance errors. Journal of Econometrics 104, 49–65.CrossRefGoogle Scholar
Jakubowski, A. (1997) A non-Skorohod topology on Okhotsk space. Electronic Journal of Probability 2, 1–21.CrossRefGoogle Scholar
Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207–248.CrossRefGoogle Scholar
Knight, K. (1989) Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261–278.CrossRefGoogle Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance processes. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
Lux, T. & Marchesi, M. (2000) Volatility clustering in financial markets: a microsimulation of interesting agents. International Journal of Theoretical and Applied Finance 3, 675–702.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process. Annals of Statistics 28, 1427–1451.CrossRefGoogle Scholar
Nelson, D.B. (1990) Stationary and persistence in GARCH (1, 1) models. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Pedersen, R. (2016) Targeting estimation of CCC-GARCH models with infinite fourth moments. Econometric Theory 32, 498–531.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) A shortcut to the LAD estimator asymptotics. Econometric Theory 7, 450–463.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Rachev, S.T. & Mittnik, S. (2000) Stable Paretian Models in Finance. Wiley.Google Scholar
Resnick, S. & Greenwood, P. (1979) A bivariate stable characterization and domains of attraction. Journal of Multivariate Analysis 9, 206–221.CrossRefGoogle Scholar
Samarakoon, D.M.M. & Knight, K. (2009) A note on unit root tests with infinite variance noise. Econometric Review 28, 314–334.CrossRefGoogle Scholar
Zhang, R.M. & Ling, S. (2015) Asymptotic inference for AR models with heavy-tailed G-GARCH noises. Econometric Theory 31, 880–890.CrossRefGoogle Scholar
Zhang, R.M., Sin, C.Y., & Ling, S. (2015) On functional limits of short- and long-memory linear processes with GARCH(1,1) noises. Stochastic Processes and their Applications 125, 482–512.CrossRefGoogle Scholar