Skip to main content
SpringerLink
Log in

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed driving noise. For general ARMA(p, q) equations these conditions are expressed in terms of the coefficient polynomials of the defining equations and moments of the driving noise sequence, while for p = 1 an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices.

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

  • Athanasopoulos G., Vahid F. (2008) VARMA versus VAR for macroeconomic forecasting. Journal of Business and Economic Statistics 26: 237–252

    Article  MathSciNet  Google Scholar 

  • Bougerol P., Picard N. (1992) Strict stationarity of generalized autoregressive processes. The Annals of Probability 20: 1714–1730

    Article  MathSciNet  MATH  Google Scholar 

  • Brockwell P.J., Davis R.A. (1991) Time series: Theory and methods (2nd ed.). Springer, New York, NY

    Book  Google Scholar 

  • Brockwell P.J., Lindner A. (2010) Strictly stationary solutions of autoregressive moving average equations. Biometrika 97: 765–772

    Article  MathSciNet  MATH  Google Scholar 

  • Gohberg I, Lancaster P., Rodman L. (1982) Matrix polynomials. Academic Press, New York, NY

    MATH  Google Scholar 

  • Golub G.H., van Loan C.F. (1996) Matrix computations (3rd ed.). Johns Hopkins, Baltimore, London

    MATH  Google Scholar 

  • Kailath R. (1980) Linear systems. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  • Kallenberg O. (2002) Foundations of modern probability (2nd ed.). Springer, New York, NY

    MATH  Google Scholar 

  • Klein A., Mélard G., Spreij P. (2005) On the resultant property of the Fisher information matrix of a vector ARMA process. Linear Algebra and its Applications 403: 291–313

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Statistics Department, Colorado State University, Fort Collins, CO, USA

    Peter Brockwell

  2. Statistics Department, Columbia University, New York, NY, USA

    Peter Brockwell

  3. Institut für Mathematische Stochastik, Technische Universität Braunschweig, 38106, Braunschweig, Germany

    Alexander Lindner & Bernd Vollenbröker

Authors
  1. Peter Brockwell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Lindner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernd Vollenbröker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Brockwell.

Additional information

Support from an NTH-grant of the state of Lower Saxony and from the National Science Foundation Grant DMS-1107031 is gratefully acknowledged.

About this article

Cite this article

Brockwell, P., Lindner, A. & Vollenbröker, B. Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise. Ann Inst Stat Math 64, 1089–1119 (2012). https://doi.org/10.1007/s10463-012-0357-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-012-0357-x

Keywords

Search

Navigation