THE AUSTRALIAN NATIONAL UNIVERSITY
RESEARCH SCHOOL OF FINANCE,
ACTUARIAL STUDIES AND STATISTICS
STAT4102/STAT8002
APPLIED TIME SERIES ANALYSIS
Assignment 2, 2020
Lecturer: Dr Tao Zou
Last Updated: Mon May 11 08:42:49 2020
This assignment is due at 11:00 am, Tuesday, 26 May, 2020.
This assignment is worth 15% of your final grade but is optional and redeemable.
Students are expected to complete this assignment individually. Maximum points:
30.0. You cannot get partially correct for all the questions, since each question is
usually worth 1 point. Assignments can only be submitted via Turnitin on
the Wattle. Late submission will not be accepted and the weight will roll over to
your final exam. Identical submissions are treated as cheating.
Please exactly follow the instructions of questions and write down your short
answers of the following questions in the answer sheet file on the Wattle. Note
that you do not need to copy the questions in the answer sheet. Please only submit
your finished answer sheet and do not paste any unrelated results. The data used in
this assignment can be found on the Wattle.
The significance level for all the questions is set to be 0.05.
1
Question 1 (Percentage Changes in the Unemployment Rate Con’d, 16.0
points)
We consider a time series from percentage changes in the unemployment rate of
US. The dataset is stored in “uschange.csv” on the Wattle and we only analyze the
column “Unemployment” as a time series which consists of 187 percentage changes
in the quarterly unemployment rate of US, from 1970 Q1 to 2016 Q3. The data are
from Federal Reserve Bank of St Louis (https://data.is/AnVtzB and https://data.is/
wQPcjU).
Based on the time series plot of the data “Unemployment”, we may temporarily consider
that the “Unemployment” time series is stationary. Please do not normalize the
"Unemployment" series and we will use the original first t = 1, · · · , T , T = 140
observations to perform the time series analysis. And the remaining observations
t = T + 1, · · · , T +M , M = 47 will be used to assess the forecast performance of the
modeling.
Part 1. (9 points) We want to use the original first t = 1, · · · , T , T = 140 observations
of the “Unemployment” series to build ARMA models (not pure AR or MA models).
Please answer the following questions.
a) (1 point) What are the orders of ARMA(p, q) that you determine by using the
SEACF table? Please answer this question in the answer sheet and do not give
any R codes or output about how to get this answer.
b) (1 point) Based on the orders you have determined in a), please fit this ARMA
model by using MLE. Please round your answer to four decimal places and please
do not round in the middle of the computation process. Then please paste all
the estimates of the stationary mean, AR coefficients, MA coefficients and the
variance of innovation in the answer sheet. Please do not give any R codes or
output about how to get this answer.
c) (1 point) Suppose the model you fit in b) is zt − µ = wt + φ1(zt−1 − µ) + · · ·+
φp(zt−p − µ) + θ1wt−1 + · · ·+ θqwt−q. What is the 95% confidence interval of φ1
by using the MLE? Please round your answer to four decimal places and please
do not round in the middle of the computation process. Then please write down
your answer in the answer sheet. Please do not give any R codes or output about
how to get this answer.
d) (1 point) Suppose the model you fit in b) is zt − µ = wt + φ1(zt−1 − µ) + · · ·+
φp(zt−p − µ) + θ1wt−1 + · · ·+ θqwt−q. What is the result for testing hypothesis
θ1 = 0 by using the MLE? Please write down your answer in the answer sheet.
Please do not give any R codes or output about how to get this answer.
e) (1 point) Please perform the model diagnostics to the model you fit in b). Please
paste the time series plot of the standardized residuals, the SACF plot of the
2
residuals, and the Ljung-Box test plot in the answer sheet. Please do not give
any R codes about how to get this answer in the answer sheet.
f) (1 point) Based on the model diagnostics result in e), does it indicate a good
fitting? Why or why not?
g) (1 point) What are the orders of ARMA(p, q) that you determine by using the
AIC based on the MLE? The range of candidate ARMA models we are interested
in is from ARMA(0,0), ARMA(1,0), · · ·, ARMA(10,0), ARMA(0,1), ARMA(1,1),
· · ·, ARMA(10,1), · · ·, ARMA(0,10), ARMA(1,10), · · ·, to ARMA(10,10). Please
answer this question in the answer sheet and do not give any R codes or output
about how to get this answer.
h) (1 point) Based on the orders you have determined in g), please fit this ARMA
model by using MLE. Please round your answer to four decimal places and please
do not round in the middle of the computation process. Then please paste all
the estimates of the stationary mean, AR coefficients, MA coefficients and the
variance of innovation in the answer sheet. Please do not give any R codes or
output about how to get this answer.
i) (1 point) Please paste the R codes (not R output) for all the above analyses of
Part 1 in the answer sheet.
Part 2. (7 points) Now we consider the remaining observations t = T +1, · · · , T +M ,
M = 47 to assess the forecast performance of the modeling. Please answer the
following questions.
a) (1 point) Suppose the model you fit in Part 1 b) is zt − µ = wt + φ1(zt−1 − µ) +
· · ·+ φp(zt−p − µ) + θ1wt−1 + · · ·+ θqwt−q. We assume wt iid∼ N(0, σ2w) and hence
we can obtain the 95% prediction intervals. Suppose we denote the original data
“Unemployment” by zt and denote the truncated forecasts of zT+1, · · · , zT+M
by z˜T+1|T , · · · , z˜T+M |T based on the model you fit in Part 1 b). Please plot the
times series {zt : t = 1, · · · , T +M}, and {z˜T+1|T , · · · , z˜T+M |T} with all of their
corresponding 95% prediction intervals in a same figure. Please paste the plot
that you generate in the answer sheet. Please do not give any R codes about how
to get this answer in the answer sheet.
b) (1 point) Please paste the normal Q-Q plot of the fitted residuals based on the
model you fit in Part 1 b) in the answer sheet. Please do not give any R codes
about how to get this answer in the answer sheet.
c) (1 point) Based on the result in b), do the prediction intervals in a) in this part
accurate? Why or why not?
d) (1 point) By using the truncated forecasts z˜T+1|T , · · · , z˜T+M |T , we can define the
3
square root of averaged square prediction error (SASPE)
SASPE =
√√√√ 1
M
M∑
m=1
(
zT+m − z˜T+m|T
)2
.
Based on the result of a) in this part, what is the value of the SASPE of the
truncated forecasts z˜T+1|T , · · · , z˜T+M |T ? Please round your answer to four decimal
places and please do not round in the middle of the computation process. Then
please write down your answer in the answer sheet. Please do not give any R
codes or output about how to get this answer.
e) (1 point) Please obtain the truncated forecasts of zT+1, · · · , zT+M based on the
model you fit in Part 1 h). What is the value of the SASPE of these new forecasts?
Please round your answer to four decimal places and please do not round in the
middle of the computation process. Then please write down your answer in the
answer sheet. Please do not give any R codes or output about how to get this
answer.
f) (1 point) In Part 1, we have two methods to determine the orders of ARMA,
namely the SEACF table and the AIC. Based on the results in d) and e) in this
part, which method results in a better forecast?
g) (1 point) Please paste the R codes (not R output) for all the above analyses of
Part 2 in the answer sheet.
Question 2 (ARMA(2,2), 6.0 points)
Consider an ARMA(2,2) yt = φ1yt−1 + φ2yt−2 + wt + θ1wt−1 + θ2wt−2 with wt ∼
WN(0, σ2w).
a) (1 point) What are the conditions that the coefficients θ1 and θ2 require, such
that this ARMA(2,2) is invertible? Please answer this question in the answer
sheet and do not give any mathematical details about how to get this answer.
b) (1 point) Deriving the intertible form of general ARMA(p, q) is not easy. Let us
start from the ARMA(2,2) yt above. Suppose the coefficients θ1 and θ2 satisfy
the conditions in a). As a consequence, yt has an invertible form
wt =
∞∑
j=0
pijyt−j.
Please use the method of undetermined coefficients to give the expression of pij,
j = 0, 1, 2, · · · , by using φ1, φ2, θ1 and θ2 in the answer sheet. Please do not give
any mathematical details about how to get this answer.
4
c) (1 point) Now consider a specific ARMA(2,2) with φ1 = 0.9, φ2 = −0.8, θ1 = 1/12
and θ2 = −1/12. Please use R to obtain the values of ACF ρy(h) for h = 1, · · · , 10
only, and paste those values in the answer sheet. Please do not give any R codes
about how to get this answer in the answer sheet.
d) (1 point) Still consider the specific ARMA(2,2) with φ1 = 0.9, φ2 = −0.8,
θ1 = 1/12 and θ2 = −1/12. Please use R to obtain the values of PACF ϕy(h) for
h = 1, · · · , 10 only, and paste those values in the answer sheet. Please do not
give any R codes about how to get this answer in the answer sheet.
e) (1 point) Is the ARMA(2,2) with φ1 = 0.9, φ2 = −0.8, θ1 = 1/12 and θ2 = −1/12
invertible? If the answer is no, please state the reason in the answer sheet.
If the answer is yes, please use R to obtain the values of pij weights in b) for
j = 1, · · · , 10 only, and paste those values in the answer sheet. Please do not
give any R codes about how to get this answer in the answer sheet.
f) (1 point) Please paste the R codes (not R output) related to all the above analyses
of Question 2 in the answer sheet.
Question 3 (Simulation for ARMA, 2.0 points)
a) (1 point) Consider the specific ARMA(2,2) model in Question 2 c) with σ2w = 4.
Please simulate 2,000 values from this model and paste the time series plot, the
SACF plot, the SPACF plot and the SEACF table in the answer sheet. Please
do not give any R codes about how to get this answer in the answer sheet.
b) (1 point) Please paste the R codes (not R output) for all the above analyses of
Question 3 in the answer sheet.
Question 4 (Forecast and PACF of ARMA(1,1), 2018 Final Exam Question,
6.0 points)
Consider a causal and invertible ARMA(1,1) zt − µ = φ(zt−1 − µ) + wt + θwt−1 with
wt ∼WN(0, σ2w). Suppose that this model does not have any parameter redundancy
problem and we have data z1, · · · , zT , which follow this ARMA(1,1) model.
a) (1 point) Please give the expression of the ACF ρz(h) of zt for h = 0, 1, 2, · · · , by
using φ and θ. Please simplify this expression as much as you can and please do
not give any mathematical details about how to get this answer in the answer
sheet.
b) (1 point) Please give the expression of zt’s PACF ϕz(2) by using φ and θ in the
answer sheet. Please simplify this expression as much as you can. Please do not
give any mathematical details about how to get this answer in the answer sheet.
5
c) (1 point) Please give the expression of the truncated one-step-ahead forecast of
zT+1 by using µ, φ, θ and zT , · · · , z1 in the answer sheet. Please simplify this
expression as much as you can. Please do not give any mathematical details
about how to get this answer in the answer sheet. Hint: the invertible form of
ARMA(1,1) in Topic 7 may be useful.
d) (1 point) Please give the expression of the truncated two-step-ahead forecast of
zT+2 by using µ, φ, θ and zT , · · · , z1 in the answer sheet. Please simplify this
expression as much as you can. Please do not give any mathematical details
about how to get this answer in the answer sheet. Hint: the invertible form of
ARMA(1,1) in Topic 7 may be useful.
e) (1 point) The MSPE of the truncated m-step-ahead forecast of zT+m can be
approximated when T is large based on Topic 8. Please give the expression of
this approximation by using φ, θ and σ2w in the answer sheet. Please simplify
this expression as much as you can. Please do not give any mathematical details
about how to get this answer in the answer sheet. Hint: the causal form of
ARMA(1,1) in Topic 7 may be useful.
f) (1 point) Based on the above results in d) and e), if we additionally assume
wt
iid∼ N(0, σ2w), please give the expression of the 95% prediction interval of zT+2
by using µ, φ, θ, σ2w and zT , · · · , z1 in the answer sheet. Please simplify this
expression as much as you can. Please do not give any mathematical details
about how to get this answer in the answer sheet.
6