Module Information

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.

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

Introduction
Historical background; theoretical properties of time series; the ideas of stationarity; 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.
Forecasting
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, identif-ication 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