John C., Ph.D. Brocklebank, David A. Dickey July, 2003

ISBN: 1590471822

SAS Publishing

This second edition of a 1986 publication contains additions that update this book with advances in time series forecasting. New topics include the Augmented Dickey-Fuller test, the model identification methods ESACF, SCAN and MINIC, unequal variances in time series models, and cointegration. The revisions and reorganization to chapter seven, Spectral Analysis, improve readability and comprehension. The addition of the final chapter, 'Data Mining and Forecasting', provides an introduction to the menu driven Time Series Forecasting System. SAS users who model and forecast time series data should add this book to their collection, including owners of the first edition. --Barry A. Evans, Ph.D., Manager, Forecasting GlaxoSmithKline

Preface

A time series is a set of ordered observations on a quantitative characteristic of a phenomenon at equally spaced time points. The goal of univariate time series analysis is to forecast values of a single historical series. The goal of multivariate time series analysis can be to model the relationships among component series as well as to forecast those components.

Time series analysis can be accomplished most effectively by the SAS procedures ARIMA, STATESPACE, SPECTRA, and VARMAX. To use these procedures properly, you must (1) understand the statistics you need for the analysis and (2) know how to invoke the procedures. SAS for Forecasting Time Series, Second Edition, makes it easier for you to apply these procedures to your data analysis problems.

Chapter 1,

**Overview of Time Series**, reviews the goals and key characteristics of time series. The analysis methods available through SAS/ETS software are presented, beginning with the simpler procedures FORECAST, AUTOREG, and X11 and continuing with the more powerful SPECTRA, ARIMA, and STATESPACE. This chapter shows the interrelationships among the various procedures. It ends with a discussion of linear regression, seasonality in regression, and regression with transformed data.

Chapter 2,

**Simple Models: Autoregression**, presents the statistical background necessary to model and forecast simple autoregressive (AR) processes. A three-part forecasting strategy is used with PROC ARIMA to identify, estimate, and forecast. The backshift notation is used to write a time series as a weighted sum of past shocks and to compute covariances through the Yule-Walker equations. The chapter ends with an example involving an AR process with regression techniques by overfitting.

Chapter 3,

**The General ARIMA Model**, extends the class of models to include moving averages and mixed ARMA models. Each model is introduced with its autocovariance function. Estimated autocovariances are used to determine a model to be fit, after which PROC ARIMA is used to fit the model, forecast future values, and provide forecast intervals. A section on time series identification defines the autocorrelation function, partial autocorrelation function, and inverse autocorrelation function. Newer identification techniques are also discussed. A catalog of examples is developed, and properties useful for associating different forms of these functions with the corresponding time series are described. This chapter includes the results of 150 observations generated from each of eight sample series. Stationarity and invertibility, nonstationarity, and differencing are discussed.

Chapter 4,

**The ARIMA Model: Introductory Applications**, describes the ARIMA model and its introductory applications. Seasonal modeling and model identification are explained, with Box and Jenkins’s popular airline data modeled. The chapter combines regression with time series errors to provide a richer class of forecasting models. Three cases are highlighted: Case 1 is a typical regression, case 2 is a simple transfer function, and case 3 is a general transfer function.

New in Chapter 4 for the second edition are several interesting intervention examples involving analyses of :

1. the effect on calls of charging for directory assistance

2. the effect on milk purchases of publicity about tainted milk

3. the effect on airline stock volume of the September 11, 2001, terrorist attacks.

Chapter 5,

**The ARIMA Model: Special Applications**, extends the regression with time series errors class of models to cases where the error variance can change over time—the ARCH and GARCH class. Multivariate models in which individual nonstationary series vary together over time are referred to as “cointegration” or “error correction” models. These are also discussed and illustrated.

This chapter presents new developments since the first edition of the book.

Chapter 6,

**State Space Modeling**, uses the AR model to motivate the construction of the state vector. Next, the equivalence of state space and vector ARMA models is discussed. Examples of multivariate processes and their state space equations are shown. The STATESPACE procedure is outlined, and a section on canonical correlation analysis and Akaike’s information criterion is included. The chapter ends with the analysis of a bivariate series exhibiting feedback, a characteristic that cannot be handled with the general ARIMA transfer function approach.

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