Module Content

Models for time series: Time series data, Trend, Seasonality, Cycles and Residuals, Stationary processes, Autoregressive processes (AR), Moving average processes (MA). 

Models of stationary processes: Purely indeterministic processes, ARMA processes, ARIMA processes, Estimation of the autocovariance function, Identifying an MA(q) process, Identifying an AR(p) process. 

Spectral methods: The discrete Fourier transform, The spectral density, Analysing the effects of smoothing. 

Estimation of the spectrum: The periodogram, Distribution of spectral estimates, The fast Fourier transform. 

Linear filters: The Filter Theorem, Application to autoregressive processes, Application to moving average processes, The general linear process, Filters and ARMA processes, Calculating autocovariances in ARMA models.

Estimation of trend and seasonality: Moving averages, Centred moving averages, The Slutzky-Yule effect, Exponential smoothing, Calculation of seasonal indices. 

Fitting ARIMA models: The Box-Jenkins procedure, Identification, Estimation Verification, Tests for white noise, Forecasting with ARMA models. 

State space models: Models with unobserved states, The Kalman filter, Prediction, Parameter estimation revisited. 

Statistical Packages: The use of R or Python in analysing data will run in parallel with the taught material. The emphasis is on how statistical software can be used to tackle statistical problems aligned with the syllabus. 

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. These will be complemented with sessions where students will use R or Python to work on their investigations. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials.

Assessment Methods

The module is assessed through a Portfolio of exercises and an examination.