266 Investigating Economic Trends and Cycles
05001000150020000 50 100 150 200 250Figure 6.10 The graph of 256 observations on a simulated series generated by an IMA(2, 1)
process that correspond to the sampled version of an integrated Wiener process
6.6 Decomposition of discrete-time ARIMA processes
An ARMA model can be represented by the equation:
∑pi= 0φiyt−i=∑qi= 0θiεt−i with φ 0 =θ 0 =1, (6.69)wherethas whatever range is appropriate to the analysis. To exploit the algebra of
polynomial operators, the equation can be embedded within the system:
φ(z)y(z)=θ(z)ε(z), (6.70)whereε(z)=zt{εt+εt− 1 z−^1 +···}is az-transform of the infinite white-noise
forcing function or disturbance sequence{εt−i;i=0, 1,...}and wherey(z)is the
z-transform of the corresponding data sequence. The embedded equation will be
associated withzt.
The polynomialsθ(z)andφ(z)must have all their roots outside the unit circle
to make their inverses,θ−^1 (z)andφ(z)−^1 , amenable to power series expansions
when|z|≥1. Then, it is possible to represent the system of (6.70) by the equation
y(z)=φ−^1 (z)θ(z)ε(z).
An ARIMA process represents the accumulation of the output of an ARMA pro-
cess. On defining the (backwards) difference operator∇(z)= 1 −z, thedth-order
model can be represented by:
∇d(z)α(z)y(z)=θ(z)ε(z). (6.71)The inverse of the difference operator is the summation operator∇−^1 (z)={ 1 +z+
z^2 +···}, and this might be used in representing the system of (6.71), alternatively,
by the equationy(z)=∇−d(z)α−^1 (z)θ(z)ε(z).
The difficulty here is that, if it is formed from an infinite number of inde-
pendently and identically distributed random variables, the disturbance sequence
cannot have a finite sum. For this reason, it appears that the algebra of polynomial
operators cannot be applied to the analysis of non-stationary processes.