Linear Time Series Models for Stationary data - Feweb

Linear Stochastic processes, Wold representation
An other important theoretical motivation for ARIMA modelling is the
Wold representation theorem (proof omitted):
The stochastic part of any zero-mean weakly stationary process yt
can be represented as an (MA(∞)) process:
yt =
∞
j=0
ψjεt−j,
with εt WN, where ψ0, ψ1, ψ2, . . . are parameters for which
∞
j=0
ψ 2 j < ∞, or
∞
|ψj| < ∞,
so that we obtain a finite variance and finite covariances indeed:
γk = E[ytyt−k] = E[
∞
j=0
ψjεt−j
∞
h=0
j=0
ψhεt−h−k] = σ 2 ε
∞
h=0
ψk+hψh
Chapter 7.1 Heij et al, TI Econometrics II 2006/2007 – p. 23/24