Preface xiii

Acknowledgments xvii

Acronyms xix

**1 Introduction 1**

1.1 Time Series Data 2

1.2 Random Variables and Statistical Modeling 16

1.3 Discrete-Time Models 22

1.4 Serial Dependence 22

1.5 Nonstationarity 25

1.6 Whiteness Testing 32

1.7 Parametric and Nonparametric Modeling 36

1.8 Forecasting 38

1.9 Time Series Modeling 38

1.10 Bibliographic Notes 39

Problems 39

**2 Linear Processes 43**

2.1 Definition 44

2.2 Stationarity 44

2.3 Invertibility 45

2.4 Causality 46

2.5 Representations of Linear Processes 46

2.6 Weak and Strong Dependence 49

2.7 ARMA Models 51

2.8 Autocovariance Function 56

2.9 ACF and Partial ACF Functions 57

2.10 ARFIMA Processes 64

2.11 Fractional Gaussian Noise 71

2.12 Bibliographic Notes 72

Problems 72

**3 State Space Models 89**

3.1 Introduction 90

3.2 Linear Dynamical Systems 92

3.3 State space Modeling of Linear Processes 96

3.4 State Estimation 97

3.5 Exogenous Variables 113

3.6 Bibliographic Notes 114

Problems 114

**4 Spectral Analysis 121**

4.1 Time and Frequency Domains 122

4.2 Linear Filters 122

4.3 Spectral Density 123

4.4 Periodogram 125

4.5 Smoothed Periodogram 128

4.6 Examples 130

4.7 Wavelets 136

4.8 Spectral Representation 138

4.9 Time-Varying Spectrum 140

4.10 Bibliographic Notes 145

Problems 145

**5 Estimation Methods 151**

5.1 Model Building 152

5.2 Parsimony 152

5.3 Akaike and Schwartz Information Criteria 153

5.4 Estimation of the Mean 153

5.5 Estimation of Autocovariances 154

5.6 Moment Estimation 155

5.7 Maximum-Likelihood Estimation 156

5.8 Whittle Estimation 157

5.9 State Space Estimation 160

5.10 Estimation of Long-Memory Processes 161

5.11 Numerical Experiments 178

5.12 Bayesian Estimation 180

5.13 Statistical Inference 184

5.14 Illustrations 189

5.15 Bibliographic Notes 193

Problems 194

**6 Nonlinear Time Series 209**

6.1 Introduction 210

6.2 Testing for Linearity 211

6.3 Heteroskedastic Data 212

6.4 ARCH Models 213

6.5 GARCH Models 216

6.6 ARFIMA-GARCH Models 218

6.7 ARCH(1) Models 220

6.8 APARCH Models 222

6.9 Stochastic Volatility 222

6.10 Numerical Experiments 223

6.11 Data Applications 225

6.12 Value at Risk 236

6.13 Autocorrelation of Squares 241

6.14 Threshold autoregressive models 247

6.15 Bibliographic Notes 252

Problems 253

**7 Prediction 267**

7.1 Optimal Prediction 268

7.2 One-Step Ahead Predictors 268

7.3 Multistep Ahead Predictors 275

7.4 Heteroskedastic Models 276

7.5 Prediction Bands 281

7.6 Data Application 287

7.7 Bibliographic Notes 289

Problems 289

**8 Nonstationary Processes 295**

8.1 Introduction 296

8.2 Unit Root Testing 297

8.3 ARIMA Processes 298

8.4 Locally Stationary Processes 301

8.5 Structural Breaks 326

8.6 Bibliographic Notes 331

Problems 332

**9 Seasonality 337**

9.1 SARIMA Models 338

9.2 SARFIMA Models 351

9.3 GARMA Models 353

9.4 Calculation of the Asymptotic Variance 355

9.5 Autocovariance Function 355

9.6 Monte Carlo Studies 359

9.7 Illustration 362

9.8 Bibliographic Notes 364

Problems 365

**10 Time Series Regression 369**

10.1 Motivation 370

10.2 Definitions 373

10.3 Properties of the LSE 375

10.4 Properties of the BLUE 376

10.5 Estimation of the Mean 379

10.6 Polynomial Trend 382

10.7 Harmonic Regression 386

10.8 Illustration: Air Pollution Data 388

10.9 Bibliographic Notes 392

Problems 392

**11 Missing Values and Outliers 399**

11.1 Introduction 400

11.2 Likelihood Function with Missing Values 401

11.3 Effects of Missing Values on ML Estimates 405

11.4 Effects of Missing Values on Prediction 407

11.5 Interpolation of Missing Data 410

11.6 Spectral Estimation with Missing Values 418

11.7 Outliers and Intervention Analysis 421

11.8 Bibliographic Notes 434

Problems 435

**12 Non-Gaussian Time Series 441**

12.1 Data Driven Models 442

12.2 Parameter Driven Models 452

12.3 Estimation 453

12.4 Data Illustrations 466

12.5 Zero-Inflated Models 477

12.6 Bibliographic Notes 483

Problems 483

Appendix A: Complements 487

A.1 Projection Theorem 488

A.2 Wold Decomposition 490

A.3 Bibliographic Notes 497

Appendix B: Solutions to Selected Problems 499

Appendix C: Data and Codes 557

References 559

Topic Index 573

Author Index 577