# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Time Series Analysis

Unit code MATH38032 10 Level 3 Semester 2 Department of Mathematics No

### Overview

This course unit covers a variety of concepts and models useful for empirical analysis of time series data.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 1 MATH10141 Pre-Requisite Compulsory
Probability 2 MATH20701 Pre-Requisite Compulsory
Statistical Methods MATH20802 Pre-Requisite Compulsory
Introduction to Statistics MATH10282 Pre-Requisite Compulsory
Regression Analysis MATH38141 Pre-Requisite Compulsory
MATH38032 pre-requisites

Students are not permitted to take more than one of MATH38032 or MATH48032 for credit in the same or different undergraduate year.

Students are not permitted to take MATH48032 and MATH68032 for credit in an undergraduate programme and then a postgraduate programme.

Note that MATH68032 is an example of an enhanced level 3 module as it includes all the material from MATH38032

When a student has taken level 3 modules which are enhanced to produce level 6 modules on an MSc programme taken within the School of Mathematics, then they are limited to a maximum of two such modules (with no alternative arrangements available otherwise)

### Aims

To introduce the basic concepts of the analysis of time series, with emphasis on financial and economic data.

### Learning outcomes

On successful completion of this course unit students will be able to:

• Explain the concepts and general properties of stationary and integrated univariate time series.
• Explain the concepts of linear filter and linear prediction, and derive best linear predictors for time series.
• Apply the backwards shift operator and the concept of roots of the characteristic equation to the study of time series models.
• Explain the concepts of autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) and seasonal autoregressive integrated moving average (seasonal ARIMA) time series, and derive basic properties thereof.
• Apply the basic methodology of identification, estimation, diagnostic checking and model selection to time series model building.
• Explain some basic concepts in the analysis of multivariate time series - multivariate autoregressive model, joint stationarity and cointegration.

### Syllabus

• Introduction and examples of economic and financial time series, asset returns. Basic models: white noise, random walk, AR(1), MA(1). 
• Stationary time series. Autocovariance and autocorrelation functions. Linear Prediction. Yule-Walker equations.  Estimation of autocorrelation and partial autocorrelation functions. 
• Models for stationary time series - autoregressive (AR) models, moving average (MA) models, autoregressive moving average (ARMA) models. Seasonal ARMA models. Properties, estimation and model building. Diagnostic checking. 
• Non-stationary time series. Non-stationarity in variance - logarithmic and power transformations. Non-stationarity in mean. Determinisitic trends. Integrated time series. ARIMA and seasonal ARIMA models. Modelling seasonality and trend with ARIMA models. 
• Filtering, exponential smoothing, seasonal adjustments. 
• Multivariate time series. Stationarity, autocorrelation and crosscorrelation. Multivariate autoregressive model.  Markov property.  Representation of univariate autoregressive models in Markov form. 
• Model based forecasting from ARMA and ARIMA. 

### Teaching and learning methods

Three lectures and one examples class each week. In addition students should expect to spend at least six hours each week on private study for this course unit.

### Assessment methods

End of semester examination: weighting 100%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

(core) Cryer, Jonathan D and Chan, Kung-Sik. Time Series Analysis with Applications in R.  Second edition. Springer, 2008. (or a newer edition)

(essential) Cowpertwait, Paul SP and Metcalfe, Andrew V. Introductory Time Series with R. Springer, 2009.

(recommended) Shumway, Robert H and Stoffer, David S. Time Series Analysis and Its Application: With R Examples. Second edition. Springer, 2006.

(further reading) Mills, Terence C. The Econometric Modelling of Financial Time Series. Second edition. Cambridge University Press, 1999.

### Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

### Teaching staff

Staff member Role
Jingsong Yuan Unit coordinator

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

•         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
•         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
•         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
•         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.