## Description

Let at ∼ W N(0, σ2

) if not specified otherwise.

1. Let Xt be a stationary time series with mean α and autocovariances γk =

0.8

k and X¯ =

P10

t=6 Xt/5

(a) Find E(X¯).

(b) Find Var(X¯).

2. Consider the process

Zt = at + at−1 + 0.25at−2 , σ

2

a = 20.

(a) Identify the order of the ARIMA model for the process.

(b) Is {Zt} stationary?

(c) Is {Zt} invertible?

(d) Find the ACVF γ(k) and ACF ρ(k) of {Zt} for k = 0, 1, 2, 3,….

(e) Find the values of πk, k = 0, 1, 2, 3,… if the process is written as

at =

P∞

i=0 πtZt−i

.

3. Consider the AR(2) process

Zt = 0.5Zt−1 − 0.06Zt−2 + at ,

where ats are independently and identically distributed as N(0, 1).

(a) Find the roots of the AR characteristic equation.

(b) Is the process Zt stationary and causal? Why?

(c) Find the autocovariances γ(0), γ(1) and γ(2).

4. Find ACVF γ(k), k=0,1,2,3,…. of the process

Zt = 0.7Zt−4 + at .

5. Find the AR and MA representation of the process

Zt = 0.6Zt−1 + at + 0.2at−1 , at ∼ W N(0, 4).

6. Identify the following as specific ARIMA models:

a) Zt = 1.5Zt−1 − 0.5Zt−2 + at − 0.3at−1 + 0.6at−2.

b) Zt = 3Zt−1 − 3Zt−2 + Zt−3 + at + 0.1at−1.

1

7. Consider the ARMA(2,1) model

Zt = 0.6Zt−1 − 0.09Zt−2 + at − 0.2at−1 , at ∼ W N(0, 1).

a) Find the AR representation of {Zt}.

b) Find the ACF ρ(k) of {Zt} for k ∈ Z.

8. Show that for |φ| > 1,

Zt =

at

φ2

−

1 −

1

φ2

X∞

k=1

at+k

φk

.

is a white noise process with Var(Zt) = σ

2

φ2 .

2