## MAS372 Time Series

 Semester 2, 2021/22 10 Credits Lecturer: Dr Kostas Triantafyllopoulos Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Time series are observations made in time, for which the time aspect is potentially important for understanding and use. The course aims to give an introduction to modern methods of time series analysis and forecasting as applied in economics, engineering and the natural, medical and social sciences. The emphasis will be on practical techniques for data analysis, though appropriate stochastic models for time series will be introduced as necessary to give a firm basis for practical modelling. For the implementation of the methods the programming language R will be used.

Prerequisites: MAS223 (Statistical Inference and Modelling)
No other modules have this module as a prerequisite.

## Outline syllabus

• Examples of time series. Purposes of analysis. Components (trend, cycle, seasonal, irregular). Stationarity and autocorrelation.
• Approaches to time series analysis. Simple descriptive methods: smoothing, decomposition.
• Differencing. Autocorrelation. Probability models for stationary series. Autoregressive models.
• Moving average models. Partial autocorrelation. Invertibility. ARMA processes.
• ARIMA models for non-stationary series. Identification and fitting. Diagnostics. Ljung-Box statistic, introduction to forecasting.
• State space models. Filtering (Kalman filter), smoothing and forecasting.
• Trend and seasonal state space models, time-varying regression. Estimation of hyperparameters, error analysis.

## Aims

• To introduce methods to uncover structure in series of observations made through time.
• To illustrate how models for time series may be constructed and studied.
• To develop methods to analyse and forecast time series.
• To show how these methods are applied to data, and what kinds of conclusion are possible.

## Learning outcomes

• Obtain a technical understanding and appreciation of time series methods.
• Use the programming language R to apply appropriate models to real data.
• Perform calculations on forecasting and state space models, using several models.
• Obtain an appreciation of Bayesian methods in time series.
• Perform complete statistical analyses to real data and interpret the results.

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal examination. All questions compulsory.

## Full syllabus

• Chapter 1: Examples of time series, definition of time series, aims of the module and overview of the methods.
• Chapter 2: Descriptive methods for time series, using R in time series, time series plots, sample autocorrelation function, moving averages, the classical decomposition, lag plots.
• Chapter 3: Probability models for stationary time series, definition of stationarity (strong and weak stationarity), autoregressive models (AR), infinite representation of AR models, stationarity and causality of AR models, moving average models (MA), invertibility of MA models, ARMA models, autocorrelation function and causality. Non-stationary ARMA models (ARIMA).
• Chapter 4: Estimation and fitting of ARIMA models, Box-Jenkins approach, identification, fitting, maximum likelihood, diagnostics and residual analysis, model selection, examples. Forecasting, forecasting causal ARMA processes, 1-step ahead and i-step ahead forecasting, prediction intervals, forecasting non-stationary ARIMA and SARIMA processes.
• Chapter 5: State space models, motivation and definition. Filtering (Kalman filter), smoothing and forecasting (Bayesian estimation). The local level model. Use of R for computing.
• Chapter 6: Model specification and model performance. Examples of state space models: trend, seasonal and trend-seasonal state space models, time-varying regression and regression with autocorrelated errors. Estimation of hyperparameters. Error analysis and prior specification.