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.
