EF5408/8705: Topics in Econometrics, Semester B 2021/2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Problems Set 1
EF5408/8705: Topics in Econometrics, Semester B 2021/2022
1. Consider the following AR(2) process?
(1 − L + 0.21L2)yt = εt
where εt is a white noise process with Eε
= 1.
(1) Is it covariance-stationary?
(2) Compute all autocovariances.
2. Consider the following MA(2) process?
yt = (1 + 3.4L + 1.2L2)εt
where εt is a white noise process with Eε
= 1.
(1) Is it covariance-stationary?
(2) Compute all autocovariances.
(3) Is this MA process invertible?
3. Consider an AR(3) model, yt = ϕ0 + ϕ1yt − 1 + ϕ2yt −2 + ϕ3yt −3 + εt. Let (1 − ϕ1L − ϕ2L2 − ϕ3L3)− 1 = ψ(L), where ψ(L) = ∑
ψiLi .
(1) Use the method of undetermined coe
cients to 
nd the formula for ψ4 .
(2) Find the VAR(1) companion form of the AR(3) model, and compute ψ4 based on the coe
cient in the companion form and check if it is the same as the one you obtained in part
(1).
4. Consider an ARMA(1, 1) model,
yt = ϕ0 + ϕ1yt − 1 + εt + θ1εt − 1 ,
where Eε
= σ
.
(1) Find all autocovariances of this process.
(2) Suppose |ϕ1 | < 1, so that yt has an MA(∞) representation yt = ∑
ψiLiεt −i. Find ψ0,ψ1,...,ψ4 .
5. Consider the following bivariate VAR(1) process yt = Φyt − 1 + εt where
Φ = 
0.8 −0.1 
−0.2 0.6 
.
Show that this model is stationary.
6. Consider the following structural VAR models
∆rgust 
∆rgust − 1 
∆rjust ∆ygust
= Φ(L)
∆rjust − 1 ∆ygust − 1
+ Aεt ,
∆yjust 
∆yjust − 1 
where rgust is the log of the real exchange rate between Germany and the US, rjust is the log of the real exchange rate between Japan and the US, ygust is the log of German/US output ratio, yjust is the log of Japanese/US output ratio, ∆ is the 
rst di
erence operator, and Φ(L) is a matrix lag polynomial. Let εt = (εgt,εjt,εut,εwt)′ denote the structural shocks and
et = Aεt denote the reduced-form shocks, where εgt,εjt,εut , and εwt denote Germany speci
c
shock, Japan speci
c shock, US speci
c shock, and global shock, respectively. Assume that
E(εtε
) = I4 .
(a) How many additional restrictions are needed to identify matrix A in the structural VAR? (b) Suppose that a12 = a21 = a32 = a41 = 0, a13 = a23 and a33 = a43. Brie
y explain the
meaning of these restrictions. Why do you think these restrictions are reasonable to identify the country speci
c shocks and global shock?
2022-02-12