Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T04:05:55.772Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIT ROOT TESTING FOR FUNCTIONALS OF LINEAR PROCESSES

Published online by Cambridge University Press:  12 December 2005

Wei Biao Wu
Wei Biao Wu
Affiliation:
University of Chicago
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the unit root testing problem with errors being nonlinear transforms of linear processes. When the linear processes are long-range dependent, the asymptotic distributions in the unit root testing problem are shown to be functionals of Hermite processes. Functional limit theorems for nonlinear transforms of linear processes are established. The obtained results differ sharply from the classical cases where asymptotic distributions are functionals of Brownian motions.The author thanks the referee and Professor B. Hansen for their valuable suggestions. The work is supported in part by NSF grant DMS-04478704.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 1 , February 2006 , pp. 1 - 14
Copyright
© 2006 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.)

References

REFERENCES

Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.Google Scholar
Avram, F. & M. Taqqu (1987) Noncentral limit theorems and Appell polynomials. Annals of Probability 15, 767–775.Google Scholar
Baillie, R.T. (1996) Long memory processes and fractional integration in econometrics. Journal of Econometrics 73, 5–59.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Bingham, N.H., C.M. Goldie, & J.L. Teugels (1987) Regular Variation. Cambridge University Press.
Bradley, R. (1986) Basic properties of strong mixing conditions. In E. Eberlein & M.S. Taqqu (eds.), Dependence in Probability and Statistics: A Survey of Recent Results, pp. 165–192. Birkhauser.
Breuer, P. & P. Major (1983) Central limit theorems for nonlinear functionals of Gaussian fields. Journal of Multivariate Analysis 13, 425–441.Google Scholar
Caporale, G.M. & L.A. Gil-Alana (2004) Fractional co-integration and real exchange rates. Review of Financial Economics 13, 327–340.Google Scholar
Chan, N.H. & N. Terrin (1995) Inference for unstable long-memory processes with applications to fractional unit root autoregressions. Annals of Statistics 20, 1662–1683.Google Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and Its Applications 15, 487–498.Google Scholar
De Jong, R.M. & J. Davidson (2000) The functional central limit theorem and weak convergence to stochastic integrals, Part I: Weakly dependent processes. Econometric Theory 16, 621–642.Google Scholar
Dickey, D.A. & W.A. Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & W.A. Fuller (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.Google Scholar
Dittmann, I. & C.W.J. Granger (2002) Properties of nonlinear transformations of fractionally integrated processes. Journal of Econometrics 110, 113–133.Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Springer.
Doukhan, P., G. Oppenheim, & M.S. Taqqu (eds.) (2003) Theory and Applications of Long-range Dependence. Birkhauser.
Gallant, A.R. & H. White (1988) A Unified Theory for Estimation and Inference for Nonlinear Econometric Models. Blackwell.
Giraitis, L. & D. Surgailis (1986) Multivariate Appell polynomials and the central limit theorem. In E. Eberlein & M.S. Taqqu (eds.), Dependence in Probability and Statistics: A Survey of Recent Results, pp. 21–72. Birkhauser.
Gorodetskii, V.V. (1977) On the strong mixing property for linear sequences. Theory of Probability and Its Applications 22, 411–412.Google Scholar
Granger, C.W.J. & R. Joyeux (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–30.CrossRefGoogle Scholar
Granger, C.W.J. & P. Newbold (1976) Forecasting transformed series. Journal of the Royal Statistical Society, Series B 38, 189–203.Google Scholar
Hannan, E.J. (1979) The central limit theorem for time series regression. Stochastic Processes and Their Applications 9, 281–289.CrossRefGoogle Scholar
Ho, H.-C. & T. Hsing (1997) Limit theorems for functionals of moving averages. Annals of Probability 25, 1636–1669.CrossRefGoogle Scholar
Hosking, J. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Ibragimov, I.A. & Yu.V. Linnik (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff.
Mandelbrot, B. & W.J. Van Ness (1969) Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422–437.Google Scholar
Peligrad, M. (1986) Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (A survey). In E. Eberlein & M.S. Taqqu (eds.), Dependence in Probability and Statistics: A Survey of Recent Results, pp. 193–224. Birkhauser.
Pham, T.D. & L.T. Tran (1985) Some mixing properties of time series models. Stochastic Processes and Their Applications 19, 297–303.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. & Z. Xiao (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–469.CrossRefGoogle Scholar
Robinson, P.M. (2003) Time Series with Long Memory. Oxford University Press.
Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences of the USA 42, 43–47.CrossRefGoogle Scholar
Sowell, F. (1990) The fractional unit root distribution. Econometrica 58, 495–505.CrossRefGoogle Scholar
Sun, T.C. (1963) A central limit theorem for non-linear functions of a normal stationary process. Journal of Mathematical Mechanics 12, 945–978.Google Scholar
Surgailis, D. (1982) Zones of attraction of self-similar multiple integrals. Lithuanian Mathematical Journal 22, 327–340.Google Scholar
Surgailis, D. (2000) Long-range dependence and Appell rank. Annals of Probability 28, 478–497.CrossRefGoogle Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31, 287–302.CrossRefGoogle Scholar
Taqqu, M.S. (1979) Convergence of integrated processes of arbitrary Hermite rank. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 50, 53–83.CrossRefGoogle Scholar
Wang, Q.Y., Y.X. Lin, & C.M. Gulati (2002) The invariance principle for linear processes with applications. Econometric Theory 18, 119–139.CrossRefGoogle Scholar
Wang, Q.Y., Y.X. Lin, & C.M. Gulati (2003) Asymptotics for general fractionally integrated processes with applications to unit root tests. Econometric Theory 19, 143–164.Google Scholar
Withers, C.S. (1981) Conditions for linear process to be strongly mixing. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57, 477–480.CrossRefGoogle Scholar
Wu, W.B. (2002) Central limit theorems for functionals of linear processes and their applications. Statistica Sinica 12, 635–649.Google Scholar
Wu, W.B. (2003a) Empirical processes of long-memory sequences. Bernoulli 9, 809–831.Google Scholar
Wu, W.B. (2003b) Additive functionals of infinite-variance moving averages. Statistica Sinica 13, 1259–1267.Google Scholar