EMET3007 EMET8012 Assignment 1

Question 1 [8 points]: Suppose X 1 ,...,X n are iid random variables
with mean µ and finite variance σ 2 .
a) Find the expected value and variance of the sample mean
¯
X =
1
n
n
∑
i = 1
X i
b) Show that the sample average squared difference from the sample
mean
˜
S 2 =
1
n
n
∑
i = 1
( X i −
¯
X ) 2
is a biased estimator of σ 2 . i.e. show that E ˜ S 2 6= σ 2 .
c) Find an unbiased estimator for σ 2 .
d) Let X = ( X 1 ,...,X n ) 0 and let 1 n denote an n × 1 column of ones.
Show that ∑ n
i = 1 X 2 i
= X 0 X and
¯
X =
1
n 1
0
n X.
[Important for later in the course: This result relies on the X i ’s being
iid. If the X i ’s are not iid, then the result may not hold.]
Question 2 [5 points]: Write a Matlab (or other) program to generate
data from the trend-cycle model
y t = m t + c t + e t , e t ∼ N( 0,σ 2 )
m t = a 0 + a 1 e a 2 t
c t = b 1 sin ( ωt ) + b 2 cos ( ωt )
a) Plotonerealisationofthemodelwith T = 60, a 0 = 50, a 1 = 0.2, a 2 =
0.11, b 1 = 10, b 2 = 20, ω = π/6, σ 2 = 25.
b) Modify the model to have a cycle term ˜ c t where ˜ c 1 = c 1 , and for
t > 1,
˜ c
t =
m t
150
c t
Plot one realisation of the model with this new cycle term and the
parameters given in ( a ) .
c) If we wanted to model GDP, which model do you think would be
more useful/accurate and why?
Question 3 [6 points]: Write a Matlab (or similar) program to generate
data from the following mean-reverting process:
y t = µ + φ 1 y t − 1 + φ 2 y t − 2 + e t , e t ∼ N( 0,σ 2 )
for t = 1,...,T with the process initialised at y − 1 = y 0 = 5µ. Plot one
realisation of the process with T = 150, µ = 15, φ 1 = 0.2, φ 2 = 0.6, and
σ 2 = 16.
This process is reverting toward a mean. What is the mean of this pro-
cess (in theory)? How closely dies this match the sample mean from your
realised data?
Question 4 [7 points]: Suppose we wished to make an iterated two-
step-ahead forecast under the mean reverting process in Question 3. (This
question is a theory-question. Do not use the data you generated in Ques-
tion 3.)
a) Compute the (theoretical) conditional expectation E ( y T + 2 | I T ,θ ) ,
where I T istheinformationsetavailableattime T, and θ = ( µ,φ 1 ,φ 2 ) 0 .
b) In what sense, if any, is this point forecast optimal? Briefly explain.
c) Compute the (theoretical) conditional density f ( y T + 2 | I T ,θ ) .
d) In practice we cannot usually construct f ( y T + 2 | I T ,θ ) as θ is un-
known. Given this, how might we proceed in practice?
Question 5 [14 points]: Suppose we wish to compare the forecast accu-
racy of two simple methods for forecasting the Canberra inflation rate: (1)
the mean using data up to the most recent observation, and (2) a random-
walk method that uses only the most recent observation. The evaluation
Table 1: Canberra CPI Inflation rate per quarter from ABS
Quarter Inflation Quarter Inflation
2014 Q3 0.4 2018 Q2 0.4
2014 Q4 0.1 2018 Q3 0.6
2015 Q1 -0.1 2018 Q4 0.7
2015 Q2 0.4 2019 Q1 0.1
2015 Q3 0.2 2019 Q2 0.3
2015 Q4 0.2 2019 Q3 0.7
2016 Q1 0.2 2019 Q4 0.6
2016 Q2 0.2 2020 Q1 0.4
2016 Q3 0.8 2020 Q2 -2.3
2016 Q4 0.6 2020 Q3 2.3
2017 Q1 0.6 2020 Q4 0.8
2017 Q2 0.0 2021 Q1 0.9
2017 Q3 0.9 2021 Q2 0.8
2017 Q4 0.6 2021 Q3 1.3
2018 Q1 0.8 2021 Q4 1.0
period is from 2016 Q3 to 2021 Q2. For measures of accuracy, we consider
MAFE and MSFE.
a) Use the two methods (historical mean and random walk) to produce
one-step-ahead forecasts for each quarter from 2016 Q3 to 2021 Q2.
For each method, graph your forecast against the actual data.
b) Computethetwomeasuresofforecastaccuracyforthetwomethods.
Which model performs better according to these measures?
c) Find one additional measure of forecast accuracy (online or else-
where). Describe this measure, and compare the two models under
this new measure.
d) A large portion of the error in your forecasts is coming from a small
numberofoutlierdatapoints. Identifythesedatapoints, anddescribe
what real-world event is driving this anomaly.
e) Find the actual 2022 Q1 Canberra CPI inflation rate from the ABS.
Compare the realisation to your forecast.
Question 6 [7 points]: Prove the following theorem: Given a density
forecast f ( y T + h | I T ,θ ) and the absolute loss function L ( ˆ y,y ) = |
ˆ y
− y | , the
point forecast which minimises expected loss is the median m T + h . Note:
the median is the unique value m T + h such that y T + h is as equally likely
to be above m T + h as below it. Mathematically, m T + h is the unique value
which satisfies
Z
m T + h
− ∞
f ( y T + h | I T ,θ ) dy T + h =
Z
+ ∞
m T + h
f ( y T + h | I T ,θ ) dy T + h =
1
2
Clarification: The task in this question is to show that to minimise ab-
solute loss, the modeller should set ˆ y T + h = m T + h .
Question 7 [3 points]: Suppose you are an intern at a company. For a
meeting you are responsible for bringing the requisite background mate-
rial for each person, however you are unsure of exactly how many people
will attend the meeting.
a) Meeting A) You must bring copies of the agenda (a one-page sheet,
cost of $0.05) for each attendee. You are told it is vital that everyone
has a copy. Describe (qualitatively) an appropriate loss function. Is
the loss function approximately symmetric?
b) Meeting B) You need to bring copies of Mas-Colell’s Microeconomic
Theory ($100 each) to the meeting, one for each person. You are told
it would be ok if attendees share copies. Describe (qualitatively) an
appropriate loss function. Is the loss function approximately sym-
metric?
c) Having done so well providing for meetings, your company asks
you to forecast the change for the upcoming year in one of the mi-
nor inputs to the company’s production process. Describe (qualita-
tively) an appropriate loss function. Is this loss function approxi-
mately symmetric?