MATH4022-E1

The University of Nottingham

SCHOOL OF MATHEMATICAL SCIENCES

A LEVEL 4 MODULE, SPRING SEMESTER 2019-2020

TIME SERIES AND FORECASTING

Suggested time to complete: TWO Hours THIRTY Minutes

Paper set: 28/05/2020 - 10:00

Paper due: 04/06/2020 - 10:00

Answer ALL questions

Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or

typeset. Do not write close to the margins. Your solutions should include complete

explanations and all intermediate derivations. Your solutions should be based on the material

covered in the module and its prerequisites only. Any notation used should be consistent with

that in the Lecture Notes.

Guidance on the Alternative Assessment Arrangements can be found on the Faculty of Science

Moodle page: https://moodle.nottingham.ac.uk/course/view.php?id=99154#section-2

Submit your answers as a single PDF with each page in the correct orientation, to the

appropriate dropbox on the module’s Moodle page. Use the standard naming

convention for your document: [StudentID]_[ModuleCode].pdf. Please check the

box indicated on Moodle to confirm that you have read and understood the statement

on academic integrity: https://moodle.nottingham.ac.uk/pluginfile.php/6288943/mod_

tabbedcontent/tabcontent/8496/FoS%20Statement%20on%20Academic%20Integrity.pdf

A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable

and does not require magnification. Text which is not in focus or is not legible for any other

reason will be ignored. If your scan is larger than 20Mb, please see if it can easily be reduced

in size (e.g. scan in black white, use a lower dpi — but not so low that readability is

compromised).

Staff are not permitted to answer assessment or teaching queries during the assessment

period. If you spot what you think may be an error on the exam paper, note this in your

submission but answer the question as written. Where necessary, minor clarifications or

general guidance may be posted on Moodle for all students to access.

Students with approved accommodations are permitted an extension of 3 days.

The standard University of Nottingham penalty of 5% deduction per working day will

apply to any late submission.

MATH4022-E1 Turn over

MATH4022-E1

Academic Integrity in Alternative Assessments

The alternative assessment tasks for summer 2020 are to replace exams that would have

assessed your individual performance. You will work remotely on your alternative assessment

tasks and they will all be undertaken in “open book” conditions. Work submitted for

assessment should be entirely your own work. You must not collude with others or employ the

services of others to work on your assessment. As with all assessments, you also need to avoid

plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct.

They are defined in the University Academic Misconduct Policy at: https://www.nottingham.ac.

uk/academicservices/qualitymanual/assessmentandawards/academic-misconduct.aspx

Plagiarism: representing another person’s work or ideas as your own. You could do this by

failing to correctly acknowledge others’ ideas and work as sources of information in an

assignment or neglecting to use quotation marks. This also applies to the use of graphical

material, calculations etc. in that plagiarism is not limited to text-based sources. There is

further guidance about avoiding plagiarism on the University of Nottingham website.

False Authorship: where you are not the author of the work you submit. This may include

submitting the work of another student or submitting work that has been produced (in whole

or in part) by a third party such as through an essay mill website. As it is the authorship of an

assignment that is contested, there is no requirement to prove that the assignment has been

purchased for this to be classed as false authorship.

Collusion: cooperation in order to gain an unpermitted advantage. This may occur where you

have consciously collaborated on a piece of work, in part or whole, and passed it off as your

own individual effort or where you authorise another student to use your work, in part or

whole, and to submit it as their own. Note that working with one or more other students to

plan your assignment would be classed as collusion, even if you go on to complete your

assignment independently after this preparatory work. Allowing someone else to copy your

work and submit it as their own is also a form of collusion.

Statement of Academic Integrity

By submitting a piece of work for assessment you are agreeing to the following statements:

1. I confirm that I have read and understood the definitions of plagiarism, false authorship

and collusion.

2. I confirm that this assessment is my own work and is not copied from any other person’s

work (published or unpublished).

3. I confirm that I have not worked with others to complete this work.

4. I understand that plagiarism, false authorship, and collusion are academic offences and I

may be referred to the Academic Misconduct Committee if plagiarism, false authorship or

collusion is suspected.

MATH4022-E1 Turn over

1 MATH4022-E1

1. Throughout this question {} is a white noise process with variance

2

.

(a) Identify each type of ARIMA(, , ) model in the following list, stating the values of

, , and the values of the parameters of the model.

i) = 0.6−1 +

ii) = 0.125−3 + − 0.5−1

iii) = 1.5−1 − 0.5−2 + − 0.2−2

iv) = −0.75−1 + 2−2 +

where 2 = 0.** and ** are the final two digits of your Student ID number. For example

if your Student ID is 53256123 then ** is 23 and hence 2 = 0.**= 0.23.

[10 marks]

(b) State whether or not each of the models in part (a) is weakly stationary, giving your

reasons.

[10 marks]

(c) Consider an AR(2) process {} with parameters 1 = 0.6 and 2 = −0.09. Explain

why the process is weakly stationary and derive the MA(∞) representation of {}, i.e.

=

∞

∑

=0

−

where are to be determined.

[10 marks]

(d) Consider again an AR(2) process with parameters 1 = 0.6 and 2 = −0.09. Using the

Yule-Walker equations, derive an expression for the autocorrelation function (ℎ) for

ℎ > 0 and compute the correlation between the realisations of the process at lag 1.

[10 marks]

MATH4022-E1

2 MATH4022-E1

2. Throughout this question {} is a white noise process with variance

2

, and is the

backward shift operator.

(a) In the figure below there is a plot of a time series, its sample autocorrelation function

and its sample partial autocorrelation function.

i) Identify an appropriate type of ARIMA model for the time series and provide reasons

for your choice.

ii) Write down the model in the form

()( − ) = (),

where expressions for the operators () and () should be provided.

iii) Suggest approximate values for the parameters of the model, giving your reasons.

iv) How would you check if the model is a good fit?

Time

x

0 200 400 600 800

96

98

10

0

10

2

10

4

0 5 10 15 20 25 30

−

0.

2

0.

0

0.

2

0.

4

0.

6

0.

8

1.

0

Lag

AC

F

Series x

0 5 10 15 20 25 30

−

0.

4

−

0.

2

0.

0

0.

2

0.

4

Lag

Pa

rti

al

A

CF

Series x

[20 marks]

(b) Let = − 0.9−1 + 0.2−2.

What order of ARIMA model is {}, where = − −1?

Provide the conditions required for invertibility of the model for (but there is no need

to check whether or not the series is invertible).

[10 marks]

(c) An MA(1) model = (1 − 0.5)

′

has been fitted to a time series. If the residuals of

a time series look like an AR(2) of the form

(1 − 0.3 − 0.42) = ,

where {} is a white noise process, what model should be fitted next? Give your

reasons.

[10 marks]

MATH4022-E1 Turn Over

3 MATH4022-E1

3. Throughout this question {} is a white noise process with variance

2

, and is the

backward shift operator.

(a) In your own words briefly discuss the use of the Akaike and Bayesian information criteria

in fitting time series models.

[10 marks]

(b) Consider the following time series

1 = 263.1,… ,−3 = 262.8, −2 = 261.8, −1 = 262.2, = 262.1

and an AR(1) model is assumed with fitted parameters

̂ = 0.9 , ̂ = 263.0 , ̂2 = 0.4.

Obtain a prediction for the time series at three time steps ahead at time = + 3 and

provide an approximate 95% prediction interval.

[10 marks]

(c) The monthly sales of a product can be modelled as:

− = 1.1 + 0.3−12 − 0.4−24 − 0.8−1 − 0.5−24 + 0.4−25

where is a constant. State the order of the seasonal ARIMA model for = − and

identify the parameters. Furthermore show that {} is a weakly stationary process.

[10 marks]

(d) i) Derive the spectral density of {} where

(1 − )3 = .

ii) Suppose a linear filter is defined by

= − −1 + −2,

where {} is a stationary process with mean and spectral density

() =

2

2

(1 − cos).

Find the form of the transfer function and the squared gain for this filter. For what

frequencies ∈ [−, ] does the transfer function vanish?

[10 marks]

MATH4022-E1 END