Introduction to Time Series and Forecasting

Series: Springer Texts in Statistics

Brockwell, Peter J., Davis, Richard A.
2nd ed. 2002. Corr. 2nd printing, 2002, XIV, 434 p. 150 illus. with CD-ROM., Hardcover
ISBN: 0-387-95351-5

About this textbook

This book is aimed at the reader who wishes to gain a working knowledge of time series and forecasting methods as applied in economics, engineering, and the natural and social sciences. The book assumes knowledge only of basic calculus, matrix algebra and elementary statistics. This second edition contains detailed instructions on the use of the new totally windows-based computer package ITSM2000, the student version of which is included with the text. Expanded treatments are also given of several topics treated only briefly in the first edition. These include regression with time series errors, which plays an important role in forecasting and inference, and ARCH and GARCH models, which are widely used for the modeling of financial time series. These models can be fitted using the new version of ITSM. The core of the book covers stationary processes, ARMA and ARIMA processes, multivariate time series and state-space models, with an optional chapter on spectral analysis. Additional topics include the Burg and Hannan-Rissanen algorithms, unit roots, the EM algorithm, structural models, generalized state-space models with applications to time series of count data, exponential smoothing, the Holt-Winters and ARAR forecasting algorithms, transfer function models and intervention analysis. Brief introductions are also given to cointegration and to non-linear, continuous-time and long-memory models.

Table of contents

Preface    
  1    INTRODUCTION    
    1.1    Examples of Time Series    
    1.2    Objectives of Time Series Analysis    
    1.3    Some Simple Time Series Models    
    1.3.3 A General Approach to Time Series Modelling    
    1.4 Stationary Models and the Autocorrelation Function    
    1.4.1 The Sample Autocorrelation Function    
    1.4.2 A Model for the Lake Huron Data    
    1.5 Estimation and Elimination of Trend and Seasonal Components    
    1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality    
    1.5.2 Estimation and Elimination of Both Trend and Seasonality    
    1.6 Testing the Estimated Noise Sequence    
1.7 Problems  
  2    STATIONARY PROCESSES    
    2.1    Basic Properties    
    2.2    Linear Processes    
    2.3    Introduction to ARMA Processes    
    2.4 Properties of the Sample Mean and Autocorrelation Function    
    2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$    
    2.5    Forecasting Stationary Time Series    
    2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values    
    2.6    The Wold Decomposition    
1.7 Problems  
  3    ARMA MODELS    
    3.1    ARMA($p,q$) Processes    
    3.2 The ACF and PACF of an ARMA$(p,q)$ Process    
    3.2.1 Calculation of the ACVF    
    3.2.2 The Autocorrelation Function    
    3.2.3 The Partial Autocorrelation Function    
    3.3    Forecasting ARMA Processes    
1.7 Problems  
  4    SPECTRAL ANALYSIS    
    4.1    Spectral Densities    
    4.2    The Periodogram    
    4.3 Time-Invariant Linear Filters    
    4.4 The Spectral Density of an ARMA Process    
1.7 Problems  
    5 MODELLING AND PREDICTION WITH ARMA PROCESSES    
    5.1    Preliminary Estimation    
    5.1.1 Yule-Walker Estimation    
    5.1.3 The Innovations Algorithm    
    5.1.4 The Hannan-Rissanen Algorithm    
    5.2 Maximum Likelihood Estimation    
    5.3    Diagnostic Checking    
    5.3.1 The Graph of $\t=1,\ldots,n\    
    5.3.2 The Sample ACF of the Residuals    
    5.3.3 Tests for Randomness of the Residuals    
    5.4    Forecasting    
    5.5    Order Selection    
1.7 Problems  
    6 NONSTATIONARY AND SEASONAL TIME SERIES    
    6.1 ARIMA Models for Nonstationary Time Series    
    6.2    Identification Techniques    
    6.3 Unit Roots in Time Series Models    
    6.3.1 Unit Roots in Autoregressions    
    6.3.2 Unit Roots in Moving Averages    
    6.4    Forecasting ARIMA Models    
    6.5    Seasonal ARIMA Models    
    6.5.1 Forecasting SARIMA Processes    
    6.6 Regression with ARMA Errors    
1.7 Problems  
  7    MULTIVARIATE TIME SERIES    
    7.1    Examples    
    7.2 Second-Order Properties of Multivariate Time Series    
    7.3 Estimation of the Mean and Covariance Function    
    7.3.2 Estimation of $\Gamma(h)$    
    7.3.3 Testing for Independence of Two Stationary Time Series    
    7.4 Multivariate ARMA Processes    
    7.4.1 The Covariance Matrix Function of a Causal ARMA Process    
    7.5 Best Linear Predictors of Second-Order Random Vectors    
    7.6 Modelling and Forecasting with Multivariate AR Processes    
    7.6.1 Estimation for Autoregressive Processes Using Whittle's Algorithm    
    7.6.2 Forecasting Multivariate Autoregressive Processes    
    7.7    Cointegration    
1.7 Problems  
  8    STATE-SPACE MODELS    
    8.1 State-Space Representations    
    8.2    The Basic Structural Model    
    8.3 State-Space Representation of ARIMA Models    
    8.4    The Kalman Recursions    
    8.5 Estimation for State-Space Models    
    8.6 State-Space Models with Missing Observations    
    8.7    The EM Algorithm    
    8.8 Generalized State-Space Models    
1.7 Problems  
  9    FORECASTING TECHNIQUES    
    9.1    The ARAR Algorithm    
    9.1.1  Memory Shortening
    9.1.2  Fitting a Subset Autoregression    
    9.1.3  Forecasting
    9.1.4  Running the Program ARAR    
    9.2    The Holt-Winters Algorithm    
    9.3 The Holt-Winters Seasonal Algorithm
    9.4 Choosing a Forecasting Algorithm
1.7 Problems  
  10    FURTHER TOPICS    
    10.1    Transfer Function Models    
    10.1.1 Prediction Based on a Transfer-Function Model    
    10.2    Intervention Analysis    
    10.3    Nonlinear Models    
    10.3.1 Deviations From Linearity    
    10.3.2 Chaotic Deterministic Sequences    
    10.3.3 Distinguishing Between White Noise and IID Sequences    
    10.3.4 Three Useful Classes of Nonlinear Models    
    10.4    Continuous-Time Models    
    10.5    Long-Memory Models    
10.4 Problems  

APPENDIX    
  Appendix A    Random Variables    
    A.1 Distribution Functions and Expectation    
    A.2    Random Vectors    
    A.3 The Multivariate Normal Distribution    
A.3     Problems  
  Appendix B    Statistical Complements    
    B.1 Least Squares Estimation    
    B.1.1 The Gauss-Markov Theorem    
    B.1.2 Generalized Least Squares    
    B.2 Maximum Likelihood Estimation   
    B.2.1 Properties of Maximum Likelihood Estimators    
    B.3    Confidence Intervals    
    B.3.1 Large-Sample Confidence Regions    
    B.4    Hypothesis Testing    
    B.4.2 Large-Sample Tests Based on Confidence Regions    

  Appendix C    Mean Square Convergence    
    C.1    The Cauchy Criterion    

  Appendix D    An ITSM Tutorial    
    D.1    Getting Started    
    D.2 Preparing Your Data for Modelling
    D.3    Finding a Model for Your Data    
    D.4    Testing Your Model    
    D.4.3 Testing for Randomness of the Residuals    
    D.5    Prediction    
    D.6    Model Properties    
    D.6.4 Generating Realizations of a Random Series    
  Bibliography    
  Index