Skip to main content
SpringerLink
Log in

Sample autocorrelations of nonsationary fractionally integrated series

  • Articles
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

We derive the asymptotic distribution of the sample autocor-relations of nonstationary fractionally integrated processes of order d. If d≥1, the sample autocorrelations approach their probability limit one with a rate equal to the sample size. If d<1, the rate is slower and depends on d. These findings carry over to the case of detrended series. Monte Carlo evidence and an empirical example illustrate the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson, O.D. (1992): Partial Autocorrelation Properties for Non-Stationary Autoregressive Moving-Average Models; Journal of Time Series Analysis 13, 486–500.

    Article  Google Scholar 

  • Bierens, H.J. (1993): Higher-Order Sample Autocorrelations and the Unit Root Hypothesis; Journal of Econometrics 57, 137–160.

    Article  MATH  MathSciNet  Google Scholar 

  • Cheung, Y.-W. (1993): Long Memory in Foreign-Exchange Rates; Journal of Business & Economic Statistics 11, 93–101.

    Article  Google Scholar 

  • Davydov, Y.A. (1970): The Invariance Principle for Stationary Processes; Theory of Probability and Its Applications 15, 487–489.

    Article  MathSciNet  Google Scholar 

  • Dickey, D.A. and 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.

    Article  MATH  MathSciNet  Google Scholar 

  • Diebold, F.X. and G.D. Rudebusch (1989): Long Memory and Persistence in Aggregate Output; Journal of Monetary Economics 24, 189–209.

    Article  Google Scholar 

  • Diebold, F.X., S. Husted, and M. Rush (1991): Real Exchange Rates under the Gold Standard; Journal of Political Economy 99, 1252–1271.

    Article  Google Scholar 

  • Durlauf, S.N. and P.B.C. Phillips (1988): Trends versus Random Walks in Time Series Analysis; Econometrica 56, 1333–1354.

    Article  MATH  MathSciNet  Google Scholar 

  • Granger, C.W.J. and R. Joyeux (1980): An Introduction to Long-Memory Time Series Models and Fractional Differencing; Journal of Time Series Analysis 1, 15–29.

    Article  MATH  MathSciNet  Google Scholar 

  • Hamilton, J.D. (1994): Time Series Analysis; Princeton University Press.

  • Hassler, U. (1994): The Sample Autocorrelation Function of I(1) processes; Statistical Papers 35, 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  • Hassler, U. and Wolters, J. (1994): On the Power of Unit Root Tests Against Fractional Alternatives; Economics Letters 45, 1–5.

    Article  MATH  Google Scholar 

  • Hassler, U. and Wolters, J. (1995): Long Memory in Inflation Rates: International Evidence; Journal of Business & Economic Statistics 13, 37–45.

    Article  Google Scholar 

  • Hasza, D.P. (1980): The Asymptotic Distribution of the Sample Autocorrelations for an Integrated ARMA process; Journal of the American Statistical Association 75, 349–352.

    Article  MATH  MathSciNet  Google Scholar 

  • Hosking, J.R.M. (1981): Fractional Differencing; Biometrika 68, 165–176.

    Article  MATH  MathSciNet  Google Scholar 

  • Hosking, J.R.M. (1984): Asymptotic Distributions of the Sample Mean, Autocovariances and Autocorrelations of Long-Memory Time Series; Mathematics Research Center Technical Summary Report #2752, University of Wisconsin-Madison.

  • Hosking, J.R.M. (1984a): Modelling Persistence in Hydrologic Time Series using Fractional Differencing; Water Resources Research 20, 1898–1908.

    Article  Google Scholar 

  • Lo, A.W. (1991): Long-Term Memory in Stock Market Prices; Econometrica 59, 1279–1313.

    Article  MATH  MathSciNet  Google Scholar 

  • Mandelbrot, B.B. and J.W. Van Ness (1968): Fractional Brownian Motions, Fractional Brownian Noises and Applications; SIAM Review 10, 422–437.

    Article  MATH  MathSciNet  Google Scholar 

  • Newbold, P. and Ch. Agiakloglou (1993): Bias in the Sample Autocorrelations of Fractional Noise; Biometrika 80, 698–702.

    Article  MATH  Google Scholar 

  • Park, J.Y. and P.C.B. Phillips (1998): Statistical Inference in Regressions with Integrated Processes: Part I; Econometric Theory 4, 468–497.

    MathSciNet  Google Scholar 

  • Phillips, P.C.B. (1987): Time Series Regression with a Unit Root; Econometrica 55, 277–301.

    Article  MATH  MathSciNet  Google Scholar 

  • Sowell, F. (1990): The Fractional Unit Root Distribution; Econometrica 58, 495–505.

    Article  MATH  MathSciNet  Google Scholar 

  • Taqqu, M. (1975): Weak Convergence to Fractional Brownian Motion and to the Rosenblatt Process; Zeitschrift für Wahrscheinlichkeit und verwandte Gebiete 31, 287–302.

    Article  MATH  MathSciNet  Google Scholar 

  • Wichern, D.W. (1973): The Behaviour of the Sample Autocorrelation Function for an Integrated Moving Average Process; Biometrika 60, 235–239.

    Article  MATH  Google Scholar 

  • Yajima, Y. (1985): Asymptotic Properties of the Sample Autocorrelations and Partial Autocorrelations of a Multiplicative ARIMA Process; Journal of Time Series Analysis 6, 187–201.

    Article  MATH  MathSciNet  Google Scholar 

  • Yajima, Y. (1988): On Estimation of a Regression Model with Long-Memory Stationary Errors; Annals of Statistics 16, 791–807.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Statistics and Econometrics, Free University of Berlin, Boltzmannstrasse 20, 14195, Berlin, FRG

    U. Hassler

Authors
  1. U. Hassler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hassler, U. Sample autocorrelations of nonsationary fractionally integrated series. Statistical Papers 38, 43–62 (1997). https://doi.org/10.1007/BF02925214

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925214

Keywords

Search

Navigation