Module Information

Module Identifier
Module Title
Linear & Non-Linear Time Series
Academic Year
Semester 1
Mutually Exclusive
External Examiners
  • Dr Theodore Kypraios (Associate Professor - University of Nottingham)
Other Staff

Course Delivery

Delivery Type Delivery length / details
Lecture 33 x 1 Hour Lectures


Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   conventional examination  75%
Semester Assessment 2 Hours   practical project involving analyzing a given series fully  25%

Learning Outcomes

On completion of this module, students should be able to.
1. understand the ideas of autocorrelation;

2. calculate autocovariances and autocorrelations for linear time series models;

3. identify suitable models for different data sets;

4. use models to forecast future values and set confidence limits on them.

5. understand and analyse transfer function/noise models;

6. use a computer package to identify, estimate and check models relating two time series;

7. construct forecasts using transfer function/noise models;

8. recognise the need for pre-whitening in the identification of transfer functions;

9. analyse low order multivariate ARMA models;

10. recognise cointegration and understand its implications;


To introduce students to the ideas of forecasting in a mathematical context and to the area of time series modeling.
To study models for relating two or more time series, to gain practical experience of their analysis by means of a project, to introduce some concepts in non-linear time series analysis.

Brief description

Time Series Analysis has, over the past 30 years, been one of the fastest growing areas of Statistics. It is concerned with situations where data or random variables are generated sequentially through time, and this makes the variables involved dependent on one another as opposed to having independent variables as in most other Statistics problems. This module develops a class of models to cater for such dependence, and considers how they are fitted to data, as well as how they may be used to forecast future values beyond the data set.
Many of the most important uses of time series analysis concern the relationships between two or more series. The ARIMA models introduced in the module on Time Series Analysis will be extended to cater for interventions and transfer functions and to multiple ARIMA models. The final part of the course will consider the blossoming new area of non-linear analysis. Students gain experience of the methodology by undertaking a short project.


Historical background; theoretical properties of time series; the ideas of stationarity and invertibility; backward shift and difference operators
Linear Time Series Models
General linear filters. Autoregressive, Moving Average and mixed models. The ARMA family. Techniques for evaluating autocorrelation and partial autocorrelation functions. Aggregation and the case for ARMA models. Non-stationarity and ARIMA models.
Model Fitting
Identification, estimation and diagnostic checking as an iterative process. Sample autocorrelations. Least squares and conditional least squares. Differencing to achieve stationarity.
Minimum mean squared error. The Fundamental Theorem of Forecasting. Forecast error variances.
Forecasting methods
Moving averages. Exponential smoothing. Holt-Winters
Extensions of the ARIMA idea
Seasonality and SARIMA models. Time reversibility and backforecasting. Case studies.
Transfer Functions and Intervention Analysis:
Regression-autoregression models; ordinary least squares estimation and the Mann-Wald theorem. Transfer functions: interventions; impulse response function; stability and gain; crosscorrelation function, prewhitening, identification of transfer functions; estimation and diagnostic checking. Forecasting.
Multiple Time Series:
The multivariate ARMA model; stationarity and invertibility, marginal models, equivalent models. Cross-covariance matrices. Co-integration.
Non-Linear Time Series: an introduction

Module Skills

Skills Type Skills details
Application of Number Throughout the module.
Communication Written worksheet solutions and project report.
Improving own Learning and Performance Feedback via tutorials.
Information Technology Extensive use of a range of computer software.
Personal Development and Career planning Students exposed to an area of Statistics that has wide applicability.
Problem solving Problem solving is central to the development and fitting of time series models, and to the ultimate goal of producing accurate forecasts of future values.
Research skills Students encouraged to consult relevant literature and compare various methods.
Subject Specific Skills General modeling ability.
Team work N/A


This module is at CQFW Level 7