Question 1

Consider the second-order autoregressive process

yt = 0 + 2yt−2 + t

where |2| < 1, and t WN(0, 2).
Derive the following:
(a) Et−1yt [1 mark]
(b) Etyt+2 [1 mark]
(c) cov(yt, yt−1) [3 marks]
(d) cov(yt, yt−2) [2 marks]
Question 2
Consider the following AR(3) process:
yt = 0.4yt−1 + 0.5yt−2 − 0.3yt−3 + t
Derive the polynomial equation needed to solve for the eigenvalues of the companion matrix
for the AR(3) process above. DO NOT SOLVE THE EQUATION. [3 marks]
Question 3
Consider the following model:
yt = yt−1 + 2t + t (1)
Describe how you can make yt stationary. Also show the mathematical proof behind your
reasoning.
[3 marks]
Question 4
Let {yt} be an AR(3) process given by:
yt = 1yt−1 + 2yt−2 + 3yt−3 + t (2)
(a) Prove that
dyt =
yt−1 + 1dyt−1 + 2dyt−2 + t (3)
where
= 1 + 2 + 3 − 1 (4)
1 = −(2 + 3) (5)
2 = −3 (6)
In your workings, make sure you clearly state the additional terms that you are adding and
subtracting.
[3 marks]
(b) Derive the autoregressive polynomial associated with the time series defined in (3) and
prove that if a unit root exists, then the equation in (4) is equal to 0.
[1 mark]
Question 5
Consider the following VAR(p) model for N variables:
xt = A0 + A1xt−1 + A2xt−2 + ... + Apxt−p + t
where
t n.i.d(0,
).
If all the matrices of vector autoregressive coefficients (A1,A2, ...,Ap) are diagonal and
is
diagonal, show and explain what the VAR(p) model will essentially consist of.
[3 marks]
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