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76 Chapter 2 Stationary Processes
that ˜PnXn+h can be expressed in the form
˜PnXn+h
∞
j1
αjXn+1−j,
in which case the preceding equations reduce to
∞
E Xn+h −
0, i 1, 2,...,
or equivalently,
j1
αjXn+1−j
Xn+1−i
∞
γ(i− j)αj γ(h+ i − 1), i 1, 2,....
j1
This is an infinite set of linear equations for the unknown coefficients αi that determine
˜PnXn+h, provided that the resulting series converges.
Properties of ˜Pn:
Suppose that EU2 < ∞, EV 2 < ∞, a,b, and c are constants, and Ɣ Cov(W, W).
1. E[(U − ˜Pn(U))Xj] 0,j ≤ n.
2. ˜Pn(aU + bV + c) a ˜Pn(U) + b ˜Pn(V ) + c.
3. ˜Pn(U) U if U is a limit of linear combinations of Xj, j ≤ n.
4. ˜Pn(U) EU if Cov(U, Xj) 0 for all j ≤ n.
These properties can sometimes be used to simplify the calculation of
˜PnXn+h, notably when the process {Xt} is an ARMA process.
Example 2.5.7 Consider the causal invertible ARMA(1,1) process {Xt} defined by
Xt − φXt−1 Zt + θZt−1, {Zt} ∼WN 0,σ 2 .
We know from (2.3.3) and (2.3.5) that we have the representations
Xn+1 Zn+1 + (φ + θ)
and
Zn+1 Xn+1 − (φ + θ)
∞
j1
∞
j1
φ j−1 Zn+1−j
(−θ) j−1 Xn+1−j.