Exercise 6.9.7. Let et be a white noise process. The spectral density of et is fe(ν) =σ 2; see Exercise 6.2. Let |φ1| < 1 and |θ1| < 1.

(a) Find the spectral density of yt = et −θ1et−1 .

Sketch the graph of the spectral density for θ1 = .5 and θ1 = 1. Take σ 2 = 1. Which frequencies are most important in yt?

(b) Show that the spectral density of yt =φ1yt−1+et is fy(ν) =

σ 2 1−2φ1 cos(2πν)+φ 2 1

.

Sketch the graph of the spectral density for φ1 = .5 and σ 2 = 1. Which frequencies are most important in yt?

(c) Find the spectral density of yt =φ1yt−1+et −θ1et−1 .

Sketch the graph of the spectral density for φ1 = .5, θ1 = .5, and σ 2 = 1. Which frequencies are most important in yt?

(d) Find the spectral density of the symmetric moving average of order 5, wt =

1 5

(et−2+et−1+et +et+1+et+2) .

Sketch the graph of the spectral density for σ 2 = 1. Which frequencies are most important in wt?