Introductory description

Much real data is collected over time reflecting snapshots of the state of an evolving system. This data forms time series. The statistical modelling of time series data is of widespread importance, for example in modelling financial data, traffic flows, biological systems and the motion of celestial bodies to name but a few areas. This module aims to provide the relevant statistical theory and experience in modelling, and performing inference for, time series data. Examples will be drawn from areas including, but not limited to, finance.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Time series: stochastic processes with discrete index set. Examples thereof. Exploratory data analysis for time series. Smoothing, e.g. Savitzky-Golay, and the Slutzky-Yule effect.
2. Notions of stationarity: strict and second order. Autocorrelation and partial autocorrelation.
3. Level, trend and seasonality. Price and return data as exemplars (including high-frequency data). Directionality.
4. Linear models of time series. AR, MA, ARMA Models and selected extensions if time permits. Forecasting.
5. Model selection; additional considerations with AIC in time-series contexts.
6. Nonlinear modelling; ARCH and GARCH models. Combining ARMA and GARCH models and the need for still greater flexibility.
7. Hidden Markov models; state space models; switching state space models. The concepts of filtering, smoothing, prediction and forecasting as well as parameter inference and model selection in this context.
8. State-space representations of ARMA models.
9. Extended example of state space modelling: Stochastic volatility. Switching stochastic volatility.

Learning outcomes

By the end of the module, students should be able to:

• Evaluate and apply techniques of time series analysis.
• Initiate an exploratory and then create a descriptive analysis of time series data, with reference to applications including those in finance.
• Derive statistical properties of linear and nonlinear time series models.
• Apply nonlinearity in time series modelling in a variety of situations.
• Create modelling studies of time series involving forecasting and simulation covering model choice, fitting and validation.

Indicative reading list

Reading lists can be found in Talis

Subject specific skills

• Demonstrate facility with rigorous  probabilistic methods.

• Evaluate, select and apply appropriate mathematical and/or probabilist techniques.

• Demonstrate knowledge of and facility with formal probability concepts, both explicitly and by applying them to the solution of finance problems.

• Create structured and coherent arguments communicating them in written form. 

• Construct logical mathematical arguments with clear identification of assumptions and conclusions.

• Reason critically, carefully, and logically and derive (prove) mathematical results.

Transferable skills

• Problem solving: Use rational and logical reasoning to deduce appropriate and well-reasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

• Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

• Communication: Present arguments, knowledge and ideas, in a range of formats.

• Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Self-motivated, setting and achieving goals, prioritising tasks.