## Description

1. Consider the following AR(3) process:

Xt =

2

5

Xt−1 +

1

4

Xt−2 −

1

10

Xt−3 + Zt

, {Zt}t∈N ∼ W N(0, σ2

).

(a) Show that {Xt}t∈N is stationary.

(b) Using the Yule-Walker equations, derive the autocorrelation function for {Xt}t∈N. Show

all steps.

(c) Assuming Zt ∼ N (0, 1.96), use the arima.sim() function to simulate 2000 observations

from the process {Xt}t∈N and plot the sample acf for the first 10 lags along with the

theoretical acf from part (b).

Use set.seed(123) before simulating the process. Include your R code.

2. Consider the following ARMA(1,1) process:

Xt =

7

10

Xt−1 + Zt −

1

10

Zt−1, {Zt}t∈N ∼ W N(0, σ2

).

(a) Check whether the process is stationary and invertible.

(b) Write the above ARMA(1, 1) process as a pure MA process.

(c) Write the above ARMA(1, 1) process as a pure AR process.

(d) Derive the autocorrelation function for {Xt}t∈N. Show all steps.

3. Show that a SARIMA(2, 1, 0) × (0, 1, 2)12 process can be written as an ARMA(p, q) process.

Specify the values of p and q.