Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
AR and MA representation of partial autocorrelation functions, with applications
Download PDF
Download PDF
  • Published: 27 April 2007

AR and MA representation of partial autocorrelation functions, with applications

  • Akihiko Inoue1 

Probability Theory and Related Fields volume 140, pages 523–551 (2008)Cite this article

  • 255 Accesses

  • 15 Citations

  • Metrics details

Abstract

We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Baillie R.T. (1996). Long memory processes and fractional integration in econometrics. J. Econometrics 73: 5–59

    Article  MATH  MathSciNet  Google Scholar 

  2. Barndorff-Nielsen O. and Schou G. (1973). On the parametrization of autoregressive models by partial autocorrelations. J. Multivariate Anal. 3: 408–419

    Article  MathSciNet  MATH  Google Scholar 

  3. Baxter G. (1961). A convergence equivalence related to polynomials orthogonal on the unit circle. Trans. Amer. Math. Soc. 99: 471–487

    Article  MATH  MathSciNet  Google Scholar 

  4. Baxter G. (1962). An asymptotic result for the finite predictor. Math Scand. 10: 137–144

    MATH  MathSciNet  Google Scholar 

  5. Beran J. (1994). Statistics for Long-memory Processes. Chapman and Hall, New York

    MATH  Google Scholar 

  6. Berk K.N. (1974). Consistent autoregressive spectral estimates. Ann. Statist. 2: 489–502

    MATH  MathSciNet  Google Scholar 

  7. Bingham N.H., Goldie C.M. and Teugels J.L. (1989). Regular Variation, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  8. Bloomfield P., Jewell N.P. and Hayashi E. (1983). Characterization of completely nondeterministic stochastic processes. Pacific J. Math. 107: 307–317

    MATH  MathSciNet  Google Scholar 

  9. Brockwell P.J. and Davis R.A. (1991). Time Series : Theory and Methods, 2nd edn. Springer, New York

    Google Scholar 

  10. Damanik D. and Killip R. (2004). Half-line Schrödinger operators with no bound states. Acta Math. 193: 31–72

    Article  MATH  MathSciNet  Google Scholar 

  11. Dégerine S. (1990). Canonical partial autocorrelation function of a multivariate time series. Ann. Statist. 18: 961–971

    MATH  MathSciNet  Google Scholar 

  12. Dégerine S. and Lambert-Lacroix S. (2003). Characterization of the partial autocorrelation function of nonstationary time series. J. Multivariate Anal. 87: 46–59

    Article  MATH  MathSciNet  Google Scholar 

  13. Durbin J. (1960). The fitting of time series model. Rev. Inst. Int. Stat. 28: 233–243

    Article  MATH  Google Scholar 

  14. Dym H. (1978). A problem in trigonometric approximation theory. Illinois J. Math. 22: 402–403

    MATH  MathSciNet  Google Scholar 

  15. Golinskii B.L. and Ibragimov I.A. (1971). A limit theorem of G. Szego. Izv. Akad. Nauk SSSR Ser. Mat. 35: 408–427 [Russian]

    MathSciNet  Google Scholar 

  16. Granger C.W. and Joyeux R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1: 15–29

    MATH  MathSciNet  Google Scholar 

  17. Grenander U. and Szegö G. (1958). Toeplitz Forms and Their Applications. University California Press, Berkeley-Los Angeles

    MATH  Google Scholar 

  18. Hardy G.H., Littlewood J.E. and Pólya G. (1952). Inequalities, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  19. Hegerfeldt G.C. (1974). From Euclidean to relativistic fields and on the notion of Markoff fields. Comm. Math. Phys. 35: 155–171

    Article  MATH  MathSciNet  Google Scholar 

  20. Hosking J.R.M. (1981). Fractional differencing. Biometrika 68: 165–176

    Article  MATH  MathSciNet  Google Scholar 

  21. Ibragimov I.A. and Rozanov Y.A. (1978). Gaussian Random Processes. Springer, New York

    MATH  Google Scholar 

  22. Ibragimov I.A. and Solev V.N. (1968). Asymptotic behavior of the prediction error of a stationary sequence with a spectral density of special type. Theory Probab. Appl. 13: 703–707

    Article  MathSciNet  MATH  Google Scholar 

  23. Inoue A. (1997). Regularly varying correlation functions and KMO-Langevin equations. Hokkaido Math. J. 26: 1–26

    MathSciNet  Google Scholar 

  24. Inoue A. (2000). Asymptotics for the partial autocorrelation function of a stationary process. J. Anal. Math. 81: 65–109

    Article  MATH  MathSciNet  Google Scholar 

  25. Inoue A. (2002). Asymptotic behavior for partial autocorrelation functions of fractional ARIMA processes. Ann. Appl. Probab. 12: 1471–1491

    Article  MATH  MathSciNet  Google Scholar 

  26. Inoue A. and Kasahara Y. (2004). Partial autocorrelation functions of fractional ARIMA processes with negative degree of differencing. J. Multivariate Anal. 89: 135–147

    Article  MATH  MathSciNet  Google Scholar 

  27. Inoue A. and Kasahara Y. (2006). Explicit representation of finite predictor coefficients and its applications. Ann. Statist. 34: 973–993

    Article  MATH  MathSciNet  Google Scholar 

  28. Inoue A. and Nakano Y. (2007). Optimal long-term investment model with memory. Appl. Math. Optim. 55: 93–122

    Article  MATH  MathSciNet  Google Scholar 

  29. Kokoszka P.S. and Taqqu M.S. (1995). Fractional ARIMA with stable innovations. Stochast. Process. Appl. 60: 19–47

    Article  MATH  MathSciNet  Google Scholar 

  30. Levinson N. (1947). The Wiener RMS (root-mean square) error criterion in filter design and prediction. J. Math. Phys. Mass. Inst. Tech. 25: 261–278

    MathSciNet  Google Scholar 

  31. Levinson N. and McKean H.P. (1964). Weighted trigonometrical approximation on R 1 with application to the germ field of a stationary Gaussian noise. Acta Math. 112: 98–143

    Article  MathSciNet  Google Scholar 

  32. Okabe Y. (1986). On KMO-Langevin equations for stationary Gaussian processes with T-positivity. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33: 1–56

    MATH  MathSciNet  Google Scholar 

  33. Osterwalder K. and Schrader R. (1973). Axioms for Euclidean Green’s functions. Comm. Math. Phys. 31: 83–112

    Article  MATH  MathSciNet  Google Scholar 

  34. Pourahmadi M. (2001). Foundations of Time Series Analysis and Prediction Theory. Wiley-Interscience, New York

    MATH  Google Scholar 

  35. Ramsey F.L. (1974). Characterization of the partial autocorrelation function. Ann. Statist. 2: 1296–1301

    MATH  MathSciNet  Google Scholar 

  36. Rozanov Y.A. (1967). Stationary Random Processes. Holden-Day, San Francisco

    MATH  Google Scholar 

  37. Rudin W. (1991). Functional Analysis, 2nd edn. McGraw–Hill, New York

    MATH  Google Scholar 

  38. Seghier A. (1978). Prediction d’un processus stationnaire du second ordre de covariance connue sur un intervalle fini. Illinois J. Math. 22: 389–401

    MATH  MathSciNet  Google Scholar 

  39. Simon, B.: The Golinskii–Ibragimov method and a theorem of Damanik and Killip. Int. Math. Res. Not. 1973–1986 (2003)

  40. Simon B. (2005). OPUC on one foot. Bull. Amer. Math. Soc. (N.S.) 42: 431–460

    Article  MATH  MathSciNet  Google Scholar 

  41. Simon B. (2005). Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory. American Mathematical Society, Providence

    MATH  Google Scholar 

  42. Simon B. (2005). Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory. American Mathematical Society, Providence

    MATH  Google Scholar 

  43. Szegö, G.: Orthogonal Polynomials, American Mathematical Society, Providence (1939) (3rd ed. 1967)

  44. Verblunsky S. (1935). On positive harmonic functions: A contribution to the algebra of Fourier series. Proc. Lond. Math. Soc. 38(2): 125–157

    Article  Google Scholar 

  45. Verblunsky S. (1936). On positive harmonic functions (second paper). Proc. Lond. Math. Soc. 40(2): 290–320

    Article  Google Scholar 

  46. Wu W.B. (2002). Central limit theorems for functionals of linear processes and their applications. Statist. Sinica 12: 635–649

    MATH  MathSciNet  Google Scholar 

  47. Zygmund A. (2002). Trigonometric Series. vol. II. 3rd edn. Cambridge University Press, Cambridge

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, 060-0810, Japan

    Akihiko Inoue

Authors
  1. Akihiko Inoue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akihiko Inoue.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Inoue, A. AR and MA representation of partial autocorrelation functions, with applications. Probab. Theory Relat. Fields 140, 523–551 (2008). https://doi.org/10.1007/s00440-007-0074-1

Download citation

  • Received: 05 August 2001

  • Revised: 28 March 2007

  • Published: 27 April 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0074-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Partial autocorrelation functions
  • Verblunsky coefficients
  • Orthogonal polynomials on the unit circle
  • Baxter’s condition
  • Fractional ARIMA processes
  • Long memory

Mathematics Subject Classification (2000)

  • Primary: 62M10
  • Secondary: 42C05
  • Secondary: 60G10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature