TMS088/MSA410, Financial time series, 2018/19

Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

The course representatives for the course are: Robert Ellis (TKIEK), Stefan Eng (GU), Alfred Lindén (TKIEK), Nastaran Soltanipour (MPCAS), Linus Wiskman (MPENM).

8/1: The exam tasks and solution to the January re-exam are now online.
2/9: The first reexam is now marked and everybody should have received the result in Ladok. Recall that bonus points do not count for this exam and that you need 30 points to pass (GU: 30 for G, 45 for VG; Chalmers: 30 for 3, 40 for 4, 50 for 5). The next re-exam is scheduled for January 3 2020 at 08.30 in the morning.
27/6: The exam is now marked and everybody should have received the result in Ladok. Recall that you need 30 points (with bonus) to pass (GU: 30 for G, 45 for VG; Chalmers: 30 for 3, 40 for 4, 50 for 5). The re-exam is scheduled for August 22 at 08.30 in the morning.
14/6: The exam tasks and solution are now online. NB: There was a slight typo in the exam that has now been corrected. x1 =/= x_2 in Task 4 c) was supposed to say x_1^2 =/= x_2^2. The correction of this part of the exam was done with this typo in mind.
29/5: The final version of the lecture notes are now online. Comments are still greatly appreciated.
28/5: Note the different times and places of tomorrow's lecture (10.00-11.45 in Euler) and exercise session (13.15-15.00 in MVH12).
22/5: In Task 5 on Project 2, you were asked to compute the residuals for t = 1000, 1002, ..., 1250. It should say t = 1000, 1001, ..., 1250. In the project description for Project 2, it said that the infer method returned residuals. That is not the case. The project description has been updated. Errors resulting from these mistakes in the project description will not be penalized.
20/5: In Task 5 on Project 2, you should write down the distribution of X_{t+1} given (X_s, s \le t) and not given (X_s, s = 1, ..., t), i.e., \sigma_{t+1} can be treated as a real number in the derivation of the confidence interval (although you should explain why!).
3/5: Project 2 is now online. Note! The deadline has been extended to May 22nd.
1/5: The corrected reports for Project 1 have now been sent out to the e-mail adresses you used to hand them in.
22/4: Apparently the hallway at MVL2085 is closed during Easter Monday. Therefore, it's fine to hand in your handwritten original for the last task of Project 1 to Andreas at the first lecture at 13.15 May 2nd, or in the folder outside MVL2085 at any time before that.
8/4: Added a quick introduction to the multivariate normal distribution in the lecture notes.
4/4: Corrected a reference to a proposition/corollary from the lecture notes in Project 1.
21/3: Project 1 is now online. Deadline 21/4 2019.
20/3: If anyone needs to take this course and count it as TMS087, please contact Andreas as soon as possible.


Course coordinator: Andreas Petersson (

Teaching assistant: Filip Wikman (

Examiner: Annika Lang

Course literature

Main literature:

Additional texts:

You might also be interested in:

English-Swedish mathematical dictionary


The program is updated as the course progresses.


Sections Contents

Introduction to the course, to time series and stationarity
[BD] 2.1, 2.4
Characterization of stationarity
[BD] 1.6, 2.5
Hypothesis testing, forecasting time series
[BD] 2.5
Forecasting stationary time series
[BD] 1.5, 2.2
Trend and seasonality, linear processes
[BD] 3.1-3.2
ARMA processes (causality, invertibility, ACVF)
[BD] 3.1-3.2, 5.1-5.2
ARMA processes (PACF, parameter estimation)
[BD] 3.3, 5.3-5.5, 6.1, 6.3
ARMA processes (order identification, model building, forecasting, ARIMA, unit roots)
[BD] 7.1-7.2, [T] 3
(G)ARCH processes
[BD] 7.1-7.3, [T] 3
(G)ARCH processes
[T] 3.6, 4.1 [BD] 7.3
IGARCH processes, nonlinear models
[T] 4.1, 4.2
Nonparametric models, testing for nonlinearities
[T] 4.4
Nonparametric forecasting, wrapup

Recommended exercises

[BD] 1.1, 1.3*, 1.4, 1.6, 1.7 and exercises in basic probability, at least 1-4. One or two of these extra exercises will also be covered on the blackboard.
[BD] 2.1, 2.2, 2.3, 2.4, 2.7, 2.8, 2.14ab, 2.15, 2.20, 2.21
[BD] 1.10, 1.11, 1.12a, 1.13, 1.15, 3.1abcde, 3.3abcde, 3.6, 3.7, 3.8.
[BD] 3.4, 3.11, 5.3, 5.4abde, 5.8, 5.11, 5.12. Review of Project 1.
[BD] 1.8, 6.1. Exercises 1, 2, 3, in additional ARCH and GARCH exercises.
Exercises 4, 5, 6, 7 in additional ARCH and GARCH exercises. Tips on Project 2.
Non-linear models. Review of Project 2. Old exam.
Student questions. Send topic suggestions to Filip.

Exercises in bold will be covered on the blackboard during excercise sessions.
* You may assume that the process has densities associated with its finite dimensional distributions.

See also the partial answer sheet from two years ago.

Computer labs

Reference literature:

Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).

Course requirements

The learning goals of the course can be found in the course plan.


Two projects can be handed in for bonus points on the (first) ordinary exam. You are encouraged to work in groups of two, and you are to write one report per group. You may not work in groups of three or more.
Project 1: deadline 21/4. Data: exchangerate.mat.
Project 2: deadline 22/5. Data: speur3505ydaily.mat.

Do not forget to read the project instructions carefully!

General guidelines for report writing

The most important part of the bonus projects is writing a report that reads well. It is vital for anyone working in a technical field to know how to do this. You should write one complete report per group, and the report should include well-commented code. The report should preferably be written in LaTex and not exceed 10 pages, including figures but excluding code. For each figure and table you include in your report, make sure to refer to it in your text and include a caption that describes the content of the table/figure.

The report should be organised into subproblems, as the project itself. For each problem, state the task you are going to solve using your own words. Then describe how you solved the task. You should explain your understanding of the problem and your theoretical strategy on how to solve it when relevant. The implementation should also be described in your own words. This can include, for example, mentioning what MATLAB functions you used for solving the task. After this, state the result by giving resulting numbers, plots etc. Comment on your results, interpret and discuss if they are as you expected. Why or why not?

If you struggle with MATLAB, make sure to first of all consult the documentation. For instance, if you want to find out how to calculate an autocorrelation in MATLAB, google it first. Make sure you read the documentation of every function you use so you understand what it does. If you need further help, you can access the book "Learning MATLAB" by T. B. Driscoll online for free via the Chalmers Library. If you speak Swedish, you may also want to consult the course pages for our own course Programmering i MATLAB for lecture notes and other excellent resources.

If you have not used LaTeX before, the easiest way to get started is to register for an account at ShareLaTex, which is an online editor. Chalmers students can use their email address in the registration in order to automatically get a premium account. For an introduction to LaTeX, see Getting Started with LaTeX for a guide in English or LaTeX-tips by Niklas Andersson and Malin Palö for a guide in Swedish.


A written exam (7.5 hp) has to be passed with at least 50% of the overall points which is graded with VG/G/U for GU students and 5, 4, 3, U for Chalmers students. Solutions and reports for home assignments should be sent as written on the problem sheet. Solutions or reports sent later than the deadlines will not be qualified for bonus points. For the exam you are allowed to bring four pages (two sheets) of your own handwritten notes and a simple calculator. You will be able to get up to 4 bonus points per project for good solutions which are valid at the (first) ordinary exam.

Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg. Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you can do this from the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal. You will not be able to write the exam if you have not signed up.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

Old exams

Note that the definition of (G)ARCH processes were slightly different (causality was assumed) for years 2014/15 to 2017/18 compared to 2018/19.

Exam 2014/15 (first year with a written exam) and its solution
Exam 2015/16 and its solution. N.B.: In Problem 1, the assumption that Cov(X_s,Z_t) = 0 for all time points s < t is missing.
Exam 2016/17 and its solution
Exam 2017/18 and its solution
Exam 2018/19 and its solution