Introduction to Time Series and Forecasting

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