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42 Chapter 1 Introduction
1.8. Let {Zt} be IID N(0, 1) noise and define
Zt, if t is even,
Xt
(Z 2
t−1 − 1)/√2, if t is odd.
a. Show that {Xt} is WN(0, 1) but not iid(0, 1) noise.
b. Find E(Xn+1|X1,...,Xn) for n odd and n even and compare the results.
1.9. Let {x1,...,xn} be observed values of a time series at times 1,...,n, and let
ˆρ(h) be the sample ACF at lag h as in Definition 1.4.4.
a. If xt a + bt, where a and b are constants and b 0, show that for each
fixed h ≥ 1,
ˆρ(h) → 1asn →∞.
b. If xt c cos(ωt), where c and ω are constants (c 0 and ω ∈ (−π, π]),
show that for each fixed h,
ˆρ(h) → cos(ωh) as n →∞.
1.10. If mt p
k0 ckt k , t 0, ±1,..., show that ∇mt is a polynomial of degree
p − 1int and hence that ∇ p+1 mt 0.
1.11. Consider the simple moving-average filter with weights aj (2q +1) −1 , −q ≤
j ≤ q.
a. If mt c0 + c1t, show that q j−q ajmt−j mt.
b. If Zt,t 0, ±1, ±2,...,are independent random variables with mean 0 and
variance σ 2 , show that the moving average At q j−q ajZt−j is “small”
for large q in the sense that EAt 0 and Var(At) σ 2 /(2q + 1).
1.12. a. Show that a linear filter {aj} passes an arbitrary polynomial of degree k
without distortion, i.e., that
mt
j
ajmt−j
for all kth-degree polynomials mt c0 + c1t +···+ckt k , if and only if
⎧
aj ⎪⎨
1 and
j
⎪⎩ j r aj 0, for r 1,...,k.
j
b. Deduce that the Spencer 15-point moving-average filter {aj} defined by
(1.5.6) passes arbitrary third-degree polynomial trends without distortion.