4.5 In one dimension, consider a stationary CAR model (4.31), E[Xt|X∖t] = ∑

S s = −S s ≠ 0

????s Xt−s, var[Xt|X∖t] = ????2

???? , where 1 − ????̃(????) = 1 − 2

∑S s=1 ????s coss???? ≠ 0 for all ???? ∈ [−????, ????]. Show that this process can be given a unilateral representation

∑

S s=0 dsXt−s = ????t, where ????t is a white noise process and the roots of ∑d s=0 dszs

, z ∈ ℂ, lie outside the unit disk.

Hint: Factorize P(z) = P(z−1) = 1 − ∑

S s = −S s ≠ 0

????s zs

= ∏

S i=1

(1 − ????i z)(1 − ????i z−1

), where the roots satisfy 0 ≤ |????i

| < 1, i = 1, ..., S. Then define {ds} by

∑ dszs = ∏(1 − ????i z).