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332 Chapter 10 Further Topics
are, respectively, the input and output of the transfer function model
∞
Xt2 τjXt−j,1 + Nt, (10.1.1)
j0
where T {τj,j 0, 1,...} is a causal time-invariant linear filter and {Nt} is a
zero-mean stationary process, uncorrelated with the input process {Xt1}. We further
assume that {Xt1} is a zero-mean stationary time series. Then the bivariate process
{(Xt1,Xt2) ′ } is also stationary. Multiplying each side of (10.1.1) by Xt−k,1 and then
taking expectations gives the equation
∞
γ21(k) τjγ11(k − j). (10.1.2)
j0
Equation (10.1.2) simplifies a great deal if the input process happens to be white
noise. For example, if {Xt1} ∼WN(0,σ2 1 ), then we can immediately identify tk from
(10.1.2) as
τk γ21(k)/σ 2
1 . (10.1.3)
This observation suggests that “prewhitening” of the input process might simplify the
identification of an appropriate transfer function model and at the same time provide
simple preliminary estimates of the coefficients tk.
If {Xt1} can be represented as an invertible ARMA(p, q) process
φ(B)Xt1 θ(B)Zt, {Zt} ∼WN 0,σ 2
Z , (10.1.4)
then application of the filter π(B) φ(B)θ−1 (B) to {Xt1} will produce the whitened
series {Zt}. Now applying the operator π(B) to each side of (10.1.1) and letting
Yt π(B)Xt2, we obtain the relation
where
Yt
∞
j0
N ′
t π(B)Nt,
τjZt−j + N ′
t ,
and {N ′
t } is a zero-mean stationary process, uncorrelated with {Zt}. The same arguments
that led to (10.1.3) therefore yield the equation
τj ρYZ(j)σY /σZ, (10.1.5)
where ρYZ is the cross-correlation function of {Yt} and {Zt}, σ 2 Z Var(Zt), and
σ 2 Y Var(Yt).
Given the observations {(Xt1,Xt2) ′ ,t 1,...,n}, the results of the previous
paragraph suggest the following procedure for estimating {τj} and analyzing the
noise {Nt} in the model (10.1.1):