Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T02:24:06.921Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTICS FOR GENERAL FRACTIONALLY INTEGRATED PROCESSES WITH APPLICATIONS TO UNIT ROOT TESTS

Published online by Cambridge University Press:  08 January 2003

Qiying Wang ,
Yan-Xia Lin  and
Chandra M. Gulati
Qiying Wang
Affiliation:
Australian National University and University of Wollongong
Yan-Xia Lin
Affiliation:
University of Wollongong
Chandra M. Gulati
Affiliation:
University of Wollongong
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, functional limit theorems for general fractional processes are established under quite weak conditions. The results are then used to derive weak convergence of general nonstationary fractionally integrated processes and to characterize unit root distribution in a model with error being a fractional autoregressive moving average process or a nonstationary fractionally integrated process.The authors thank three referees and an associate editor for their detailed reading of this paper and valuable comments, which have led to this much improved version of the paper.

Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 1 , February 2003 , pp. 143 - 164
Copyright
© 2003 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

Avram, F. & M.S. 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
Beran, J. (1994) Statistics for Long Memory Processes. New York: Chapman and Hall.
Beran, J. (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory ARIMA models. Journal of the Royal Statistical Society, Series B 57, 695–672.Google Scholar
Beran, J., R.J. Bhansali, & D. Ocker (1998) On unified model selection for stationary and nonstationary short and long memory autoregressive processes. Biometrika 85, 921–934.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
Brockwell, P.J. & R.A. Davis (1987) Time Series: Theory and Methods. New York: Springer-Verlag.
Chan, N.H. & N. Terrin (1995) Inference for unstable long-memory processes with applications to fractional unit root autoregressions. Annals of Statistics 23, 1662–1683.Google Scholar
Chan, N.H. & B.S. Tsay (1996) Asymptotic inference for non-invertible moving-average time series. Journal of Time Series Analysis 17, 1–17.Google Scholar
Chan, N.H. & C.Z. Wei (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.Google Scholar
Chow, Y.S. (1965) Local convergence of martingales and the law of large numbers. Annals of Mathematics and Statistics 36, 552–558.Google Scholar
Csörgõ, M. & J. Horváth (1993) Weighted Approximation in Probability and Statistics. Wiley Series in Probability and Mathematical Statistics: Probability and Statistics. Chichester: Wiley.
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.Google Scholar
Davidson, J. & R.M. Jong (2000) The functional central limit theorem and weak convergence to stochastic integrals, II: Fractionally integrated processes. Econometric Theory 16, 643–666.Google Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and Its Applications 15, 487–498.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
Fox, R. & M.S. Taqqu (1986) Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 517–532.Google Scholar
Giraitis, L. & D. Surgailis (1990) A central limit theorem for quadratic forms in strongly dependent linear variables and its applications to the asymptotic normality of Whittle's estimate. Probability Theory and Related Fields 86, 87–104.Google Scholar
Gorodetskii, V.V. (1977) On convergence to semi-stable Gaussian process. Theory of Probability and Its Applications 22, 498–508.Google Scholar
Granger, C.W.J. & A. Joyeux (1980) An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis 1, 15–29.Google Scholar
Hall, A.R. (1989) Testing for a unit root in the presence of moving average errors. Biometrika 76, 49–56.Google Scholar
Hall, P. & C.C. Heyde (1980) Martingale Limit Theory and its Applications. New York: Academic Press.
Hannan, E.J. (1979) The central limit theorem for time series regression. Stochastic Processes and Their Applications 9, 281–289.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.Google Scholar
Hosking, J.R.M. (1984) Modeling persistence in hydrological time series using fractional differencing. Water Resources Research 20, 1898–1907.Google Scholar
Kokoszka, P.S. & M.S. Taqqu (1995) Fractional ARIMA with stable innovations. Stochastic Processes and Their Applications 60, 19–47.Google Scholar
Komlós, J., P. Major, & G. Tusnády (1975) An approximation of partial sums of independent R.V.'s and sample DF, part I. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 32, 111–131.Google Scholar
Komlós, J., P. Major, & G. Tusnády (1976) An approximation of partial sums of independent R.V.'s and sample DF, part II. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 34, 33–58.Google Scholar
Li, W.K. & A.I. McLeod (1986) Fractional time series modeling. Biometrika 73, 217–221.Google Scholar
Liu, M. (1998) Asymptotics of nonstationary fractional integrated series. Econometric Theory 14, 641–662.Google Scholar
Mandelbrot, B.B. & J.W. Van Ness (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 423–437.Google Scholar
Marinucci, D. & P.M. Robinson (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 3–13.Google Scholar
Marinucci, D. & P.M. Robinson (2000) Weak convergence of multivariate fractional processes. Stochastic Processes and Their Applications 86, 103–120.Google Scholar
Mielniczuk, J. (1997) Long and short-range dependent sums of infinite-order moving averages and regression estimation. Acta Sci. Math. (Szeged). 63, 301–316.Google Scholar
Odaki, M. (1993) On the invertibility of fractionally differenced ARIMA processes. Biometrika 80, 703–709.Google Scholar
Parks, J.Y. & P.C.B. Phillips (1988) Statistical inference in regressions with integrated processes, part 1. Econometric Theory 4, 468–497.Google Scholar
Parks, J.Y. & P.C.B. Phillips (1989) Statistical inference in regressions with integrated processes, part 2. Econometric Theory 5, 95–131.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–302.Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Phillips, P.C.B. & Z. Xiao (1998) A Primer on Unit Root Testing. Manuscript.
Robinson, P.M. (1994) Time series with strong dependence. In C.A. Sims (ed.), Advance in Econometrics, Sixth World Congress. vol. 1, pp. 47–95. Cambridge, UK: Cambridge University Press.
Robinson, P.M. (1997) Large sample inference for nonparametric regression with dependent errors. Annals of Statistics 25, 2054–2084.Google Scholar
Said, S.E. & D.A. Dickey (1984) Testing for unit roots in autoregressive moving average models of unknown order. Biometrika 71, 599–607.Google Scholar
Samorodnitsky, G. & M.S. Taqqu (1994) Stable Non-Gaussian Random Processes. New York: Chapman and Hall.
Sims, C.A., J.H. Stock, & M.W. Watson (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113–144.Google Scholar
Sowell, F.B. (1990) The fractional unit root distribution. Econometrica 58, 495–505.CrossRefGoogle Scholar
Stout, W.F. (1974) Almost Sure Convergence. New York: Academic Press.
Tanaka, K. (1999) The nonstationary fractional unit root. Econometric Theory 15, 549–582.CrossRefGoogle Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschrift für Wahrscheinlichskeitstheorie und Verwandte Gebiete 31, 287–302.CrossRefGoogle Scholar
Wang, Q., Y.-X. Lin, & C.M. Gulati (2001) Asymptotics for moving average processes with dependent innovations. Statistics and Probability Letters 54, 347–356.CrossRefGoogle Scholar
Wang, Q., Y.-X. Lin, & C.M. Gulati (2002) The invariance principle for linear processes with applications. Econometric Theory 18, 119–139.CrossRefGoogle Scholar