## Description

Let at ∼ W N(0, σ2

)

1. Does the quadratic trend Tt = α + βt2 pass through the moving average

filter (a−1, a0, a1) = ( 1

3

,

1

3

,

1

3

)?

2. Suppose Zt = 8+ 4t+ 2Xt, where Xt is a zero-mean stationary series with

autocovariance function γk.

(a) Find the mean and the autocovariance function of Zt.

(b) Is Zt stationary? Why?

(c) Find the mean and the autocovariance function of ∆Zt = (1 − B)Zt.

(d) Is ∆Zt stationary? Why?

3. Suppose that Zt = (at + at−1 + at−3)/3

(a) Show that Zt is weakly stationary.

(b) Find ρk, k = 0, 1, 2, 3, …

(c) Find Var

1

5

P5

t=1 Zt

.

4. Consider the time series {Zt} satisfying

Zt = 0.2Zt−1 + at.

(a) Assuming that {Zt} is stationary, find the mean E(Zt).

(b) Assuming that {Zt} is stationary and Cov(Zs, at) = 0 for s < t, find
the variance Var(Zt). (Hints: take variance on both sides.)
(c) Find Cov(Zt, Zt−k) for k = 1, 2, 3, ... (Hints: multiply Zt−k on both
sides and take expectation.)
5. Consider the time series {Zt} satisfying
Z1 = a1; Zt = 0.2Zt−1 + at for t > 1.

(a) By mathematical induction, show that Zt =

Pt−1

k=0 0.2

kat−k.

(b) Find the mean E(Zt) and the variance Var(Zt).

(c) Find Cov(Zt, Zt−k) for t > k and k ≥ 0.

6. Consider the data set monthly milk.csv in the class website that contains the monthly milk production from 1962 to 1975. Using R, decompose

the series into three components.

1