Wilfrid Laurier University
EC665 Take Home Final For
Chen, Zhao
Date posted on MyLS: 6 April 2021 at 8.30am
Date due: 9 April 2021 at 4.59pm EDT
Instructions
This is not a group assignment. You must complete the exam on your own. Completing
the exam in a group will be considered academic misconduct.
Appropriate steps will be taken if I suspect academic misconduct of any kind.
Complete all 4 questions.
All answers must be uploaded in a single file to MyLS in the appropriate Dropbox folder.
Your answers must include all the software code used to generate the econometric results
in an appendix. Identify clearly which code refers to which question. In addition, upload
your econometrics output files. NO other submissions will be accepted.
Questions 3 and 4 are estimations. Your personalized excel file contains all the data
necessary for the estimations. Please note that the file may contain data that you do not
need for your estimations.
The exam is worth 35% of your final grade.
There are 4 pages including the cover.
1
Question 1: (worth 6 points)
Let
[
yt
zt
]
=
[
0.8 0.2
0.2 0.8
][
yt−1
zt−1
][
e1t
e2t
]
a. Use (5.28) to find the appropriate second-order stochastic difference equation for yt.
yt =
a10(1− a22) + a12a20 + (1− a22L)e1t + a12e2t−1
(1− a11L)(1− a22L)− a12a21L2 (5.28)
b. Determine whether the {yt} sequence is stationary.
c. Discuss the shape of the impulse response function of yt to a one unit shock in e1t
and to a one unit shock in e2t.
d. Suppose e1t = εyt + 0.5εzt and e2t = εzt. Discuss the shape of the impulse response
function of yt to a one-unit shock in εyt. Repeat for a one-unit shock in εzt.
e. Suppose e1t = εyt and that e2t = 0.5εyt + εzt. Discuss the shape of the impulse
response function of yt to a one-unit shock in εyt. Repeat for a one-unit shock in
εzt.
f. Use your answers to Parts c. and d. to explain why the ordering in Choleski
decomposition is important.
g. Using the notation in (5.27), find (A1)
2 and (A1)
3. Does (A1)
n appear to approach
zero (i.e., the null matrix)?
xt = µ+
∞∑
i=0
Ai1et−i (5.27)
2
Question 2: (worth 6 points)
Consider the ARCH(2) process: Et−1(εt)2 = α0 + α1(εt−1)2 + α2(εt−2)2.
a. Suppose that the residuals come from the model yt = a0 + a1yt−1 + εt. Find the
conditional and unconditional variance of {yt} in terms of the parameters a1, α0, α1
and α2.
b. Suppose that {yt} is an ARCH-M process such that the level of {yt} is positively
related to its own conditional variance. For simplicity, let: yt = α0 + α1(εt−1)2 +
α2(εt−2)2 + εt. Trace out the impulse response function of {yt} to an εt shock. You
may assume that the system has been in long-run equilibrium (εt−2 = εt−1 = 0)
but now ε1 = 1. Thus, the issue is to find the values of y1, y2, y3, and y4 given that
ε2 = ε3 = ... = 0.
c. Use your answer to Part b to explain the following result. A student estimated {yt}
as an MA(2) process and found the residuals to be serially uncorrelated. A second
student estimated the same series as the ARCH-M process: yt = α0 + α1(εt−1)2 +
α2(εt−2)2+εt. Why might both estimates appear reasonable? How would you decide
which is the better model?
Question 3: (worth 8 points)
You expect a long run equilibrium relationship between the 3 processes ({x1t}, {x2t},
{x3t}) in the ‘Question 3’ worksheet of the excel file. Estimate an appropriate multi-
equation model and justify your econometric choices. Discuss the estimation results
and the impulse response functions.
Note that the data generating processes contain no constant and no trend.
Answers should be contained in the same file as the answers to Questions 1 to 2. Copy
the software code into an appendix.
3
Question 4: (worth 15 points)
Estimate a model of the term spread (long-term versus short-term interest rates).
You have encountered the issue in previous courses you have taken (undergraduate or
graduate). Estimate an econometric model using any one of the techniques covered in the
course EXCEPT ARIMA. It must be suitable for the question you are investigating.
Answers should be contained in the same file as the answers to Questions 1 to 3. Copy
the software code into an appendix.
There are 7 different time series for the country ‘MY - Malaysia’ in the ‘Question 4’
worksheet of the excel file. The 7 series are:
1. eq - log equity price index
2. expinf - expected inflation (%)
3. gdpyoy - GDP growth (% year over year)
4. infyoy - CPI inflation (% year over year)
5. ltir - long-term government bond yield (%)
6. reer - log real effective exchange rate
7. stir - short-term government bond yield (%)
You may ONLY use the available series. No other time series may be collected or added.
Data transformations are permitted (e.g., taking changes, etc. . . ), as are any dummy
variable you choose to create. You may not need all 7 series for your estimations.
Grading will be affected by:
• appropriateness of the technique for the problem being investigated;
• sophistication of any estimated model and variety of time series used;
• clarity of the econometrics explanations provided (point form is OK); and
• clarity of the economic explanations provided (point form is OK)
Your explanations can be backed-up by references to article(s) in peer-reviewed publica-
tions, or textbook(s). Remember that the same econometric model may not to be valid
for every country.
Good luck!!!
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