# 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
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