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Bilinear Time Series Models

  • T. Subba Rao
  • M. M. Gabr
Part of the Lecture Notes in Statistics book series (LNS, volume 24)

Abstract

In the theory of stationary random processes, Wold’s theorem (Wold, 1938) plays a fundamental role. Briefly, the theorem can be stated as follows (for a proof of the statement, see e.g. Priestley, 1981). Let Xt be a zero mean second order stationary process. Then Xt can be expressed in the form
$${{\rm{X}}_{\rm{t}}}\;{\rm{ = }}\;{{\rm{U}}_{\rm{t}}}\;{\rm{ + }}\;{{\rm{V}}_{\rm{t}}}$$
(5.1.1)
where
  1. (i)

    Ut and Vt. are uncorrected processes

     
  2. (ii)

    Ut is non-deterministic with a one-sided linear representation \(\;{{\rm{U}}_{\rm{t}}}\; = \;\sum\limits_{u = 0}^\infty {{a_u}\;{n_{t - u}}} \) with \({a_0}\; = \;1,\;\Sigma a_u^2\;{n_{t - u}}\) is an uncorrected process. The process nt is uncorrected with Vt,.i.e. E(ns Vt) = 0, all s,t. The sequences{au} and {nt} are uniquely determined.

     
  3. (iii)

    {Vt} is deterministic, i.e. can be predicted from its own past with zero prediction variance.

     

Keywords

Residual Variance Time Series Model Bilinear Model Volterra Kernel Generalise Transfer Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • T. Subba Rao
    • 1
  • M. M. Gabr
    • 2
  1. 1.Department of MathematicsUniversity of ManchesterManchesterEngland
  2. 2.Department of MathematicsUniversity of AlexandriaAlexandriaEgypt

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