Sheet - 1 Stationarity and ACF 1.1 1.3 Find Formula a. Find...

1 Stationarity and ACF
1.2 Problem 1
(MA Models) For each of the following, state if it is a stationary
process. If so, give the mean and the autocovariance functions.
Assume { W
t
} is i.i.d. N(0,1). List here:
(a) X
t
= W
t
- W
t - 3
(b) X
t
= W
3
(c) X
t
= t + W
3
(d) X
t
= W
2
t
(e) X
t
= W
t
W
t - 2
Problem 3
Find ACF for AR models, say X
t
= φX
t - 1
+ w
t
with
w
t
∼ WN (0 , σ
2
). The solution is: γ (0) = Cov ( X
t
, X
t
) =
Cov ( φX
t - 1
+ w
t - 1
, φX
t - 1
+ w
t - 1
) = φ
2
γ (0) + σ
2
w
. This solves
for γ (0) =
σ
2
w
[Solution]
1.3 Problem 2
Find
a. Find the ACF of the time series X
t
= W
t
+
5
c. Which of the MA models in a and b is invertible?
[Solution]
a. Compute three cases: since the W
t
are uncorrelated, we can
ignore any-terms of the form E W
s
W
t
when s 6 = t . Then
γ (0) = E W
2
t
+
25
1.4 Problem 3
Find ACF for AR models, say X
t
= φX
t - 1
+ w
t
with
w
t
∼ WN (0 , σ
2
). The solution is: γ (0) = Cov ( X
t
, X
t
) =
Cov ( φX
t - 1
+ w
t - 1
, φX
t - 1
+ w
t - 1
) = φ
2
γ (0) + σ
2
w
. This solves
for γ (0) =
σ
2
w