TMA4285 - Time Series Models

TMA4285 - Time Series Models


02.01.19: Exam December 2018 with solution sketch, and results


11.12.18 Eksamen 15-19 på Sluppenvegen 14: SL238 (1), SL120 (2), SL110 hvit sone (32)

09.12.18: There seem to be differences in the definitions used for the term "stationary process" (TMA4245 in year 2013 for example). We follow our textbook and have two notions: (A) Strictly stationary (Remark 1 after Def 1.4.2) and (B) Weakly stationary (Def 1.4.2). It is standard, as also described well on Wikipedia, to use the term 'stationary' to mean 'strictly stationary', but our textbook uses 'stationary' to mean 'weakly stationary'. This is unfortunate, but logical since the book almost exclusively deals with weakly stationary processes.

26.11.18: FINAL Consultance time: Monday 03.12.2018 0915-11 Room 1138, Sentralbygg 2, Gløshaugen

13.11.18: A quotation found by Google on "statistical learning from time series" : More specifically, the data I called “stock index”, was actually modeled using a random walk process. As the name indicates, a random walk is a completely stochastic process. Due to this, the idea of using historical data as a training set in order to learn the behavior and predict future outcomes is simply not possible. What is your opinion on this?

02.11.18: Dear student. Voice your opinion about the quality of your study program!

19.10.18: Exercise 8 is now posted below.

25.09.18: Notes from the first reference group meeting are found below. Links to the 2016 edition of the course textbook + more links have been added. In general it is an advantage to try to study the textbook before the lectures.

19.09.18: All future exercise classes will be in Nullrommet. If you do not have access to this room, send an email to Maria with full name and the number on your student card (bottom left number). If you do not have a user account for the computers and would like one, send an email to Maria with your full name and user name.

17.09.18: The next exercise class (19.09) will be in the computer lab Nullrommet (room number 380A) at the mathematics department. This is located on the 3rd floor in Central building 2, nordre lavblokk. There are computers there, but if you dont have a user account, please bring your own laptop. If you want a user account, please send an email to Maria. Here, you can find more information about the computer labs: ComputerLabs

11.09.18: If anyone work alone on the first project, but would like to work together with someone, or if anyone has room in their group for one more person, please contact Maria.

07.09.18: Exercise 3 . Each group will get a unique data set. Please send an email to Maria ( to obtain the data set.

31.08.18: Exercise 3 is mandatory and counts for 10% in the grading. It will be announced at Friday September 7th and the deadline for submission is Sunday September 23th (23:59). There will be no lectures in week 37. The time should be used for necessary self-study for exercise 3 which will be analysis of a data set from an ARMA process.

24.08.18: The exercise in week 36 has been moved to Friday 7. sep, 10.15-12.00 in room R60.

21.08.18: The lecture on Tuesday August 28th is cancelled.

20.08.18: Maria is away 3.-5. sep. Exercise that week will be moved to Thursday 6.sep or Friday 7.sep before noon. New time will be decided in the lecture.

10.08.18: Welcome to the course TMA4285 Time Series Models Fall 2018. The first lecture will take place in F4 Monday August 20. If any non-Scandinavian speaking students want to follow the lectures in this course, the lectures will be given in English.

Teaching Material:

Brockwell and Davis: Introduction to Time Series and Forecasting, Springer 2002 (2.Ed), 2016 (3.Ed),
(Data) (Extras) ISBN 978-0-387-95351-9,ISBN 978-3-319-29854-2

Old exams

Supplemental Reading:

GUM (2008): Bureau International des Poids et Mesures, Guide to the Expression of Uncertainty in Measurement
The ISI glossary of statistical terms in a number of languages
Brockwell and Davis: Time Series: Theory and Methods, Springer 1987, 1991 (2.edition)
Shumway and Stoffer: Time Series Analysis and Its Applications (With R Examples), Springer 2017.
Venables, Ripley (2002). Modern applied statistics with S (4th ed.). New York: Springer. Chapter 14: Time Series Analysis
Box, Jenkins, Reinsel (2008): Time Series Analysis (Forecasting and Control) (4th ed.), Wiley

Recommended preliminary courses:

TMA4267 - Linear Statistical Models (Multivariate regression)
TMA4265 - Stochastic Modeling (Stochastic processes)
TMA4145 - Linear Methods (Vector- and Hilbert space)

A more advanced related course:

MA8109 - Stochastic Processes and Differential Equations

Teachers: Gunnar Taraldsen and Maria Lie Selle

Reference group: 24.09.18: Meeting 1
12.11.18: Meeting 2 (no students present)
19.11.18: Meeting 3 (Final meeting)

Monday 08:15-10:00 F4 Gamle fysikk
Tuesday 12:15-14:00 A34 Handelshøyskolen

Wednesday 09:15-10:00 R60 Realfagbygget. Nullrommet (room number 380A), 3rd floor in Central building 2, nordre lavblokk.

Consultance time:
Monday 1015-11 Room 1138, Sentralbygg 2, Gløshaugen

Lecture Plan: Brockwell and Davis 2016

Week 34: Stochastic processes, Chap 1 and 2. App A.

Week 35: Stationary processes, Chap 2.

Week 36: Stationary processes, Chap 2.

Week 37: No lectures. Work on Exercise 3

Week 38: ARMA models, Chap 3

Week 39: Statistics for ARMA models, Chap 5

Week 40: Statistics for ARMA models, Chap 5

Week 41: Nonstationary and seasonal time series, Chap 6 and comments on Ex3 Bootstrap

Week 42: Ex3 and Ex8 discussion, Chap 9 State-Space Models, 9.1 State-Space Representations, 9.2 The Basic Structural Model, 9.3 State-Space Representation of ARIMA Models, 9.4 The Kalman Recursions, 9.5 Estimation for State-Space Models

Week 43: No lectures. Work on Excercise 8

Week 44: Time Series Models for Financial Data, Chap 7.1 Historical Overview, 7.2 GARCH Models, Chap 11.3 Nonlinear Models

Week 45: December 2008 LF Repetition

Week 46: December 2009 LF Repetition

Week 47: Main ideas in the course. Repetition. Exam questions.


Ex 1, Week 35: App A: 4,5,6, Chap 1: 2-5, Chap 2: 1. LF
Ex 2, Week 36: Chap 2: 5, 7-9, 13, 15, 18-20. LF(The order of the solutions are a bit messy, but you will find all solutions in the file.)
Ex 3, Week 37: Exercise 3, Mandatory! The deadline for submission is Sunday September 23th (23:59). Send the report by email to Maria. Remember to include the names of all group members.
Ex 4, Week 39: Chap 3: 1b, 4, 6, 7, 8, 12. LF
Ex 5, Week 40: App A: 7, Chap 5: 1,3,4,8,13 LF
Ex 6, Week 41: Chap 5: 9, 10, 11, 12.LF
Ex 7, Week 42: Chap 6:1, 2, 5, 6, 11.LF
Ex 8, Week 43: Exercise 8, DATA, Mandatory! The deadline for submission is Sunday November 4th (23:59). Send the report by email to Maria. Remember to include the names of all group members.
Ex 9, Week 45: Chap 9: 1, 3, 9, 11, 12, 17.LF
Ex 10, Week 46: Chap 7: 1, 2, 3; Chap 11: 3 LF
Week 47: Previous exams and exercises

Exam (80/100): Written, 11.12.2018, 4 hours.
Permitted examination support material: C:
– Tabeller og formler i statistikk, Tapir forlag,
– K.Rottman. Matematisk formelsamling,
– Stamped yellow A5 sheet with your own handwritten notes,
– Determined, single calculator

As for previous exams you are allowed to bring with you one yellow A5 sheet with your own formulas and notes. The sheet of paper need to be stamped and you can get it at the Department office, seventh floor SII.

Work (20/100): Two exercises are mandatory and should be handed in to Maria and count 10% each on the final grade. You can work in groups of 1-3. Use the name of all group members when you hand in the exercises, not student numbers or candidate numbers. The submission deadline and more information will be announced later.


1) Exercises

2) Lectures (in particular definition of conditional expectation and Hilbert space - which is not in the book)

3) Chapters from Brockwell and Davis 2016:
1. Introduction 2. Stationary processes 3. ARMA models 4. Spectral analysis (out) 5. Modeling and forecasting with ARMA processes 6. Nonstationary and seasonal time series models (6.1-5) 7. Time Series Models for Financial Data (7.1-2) 8. Multivariate time series (out) 9. State space models (9.1-9.5) 10. Forecasting techniques (out) 11. Further topics (only 11.3) Appendix A, B, C

2019-01-15, Hallvard Norheim Bø