You are here

Time Series Analysis

Wilfredo Palma
Publisher: 
John Wiley
Publication Date: 
2016
Number of Pages: 
579
Format: 
Hardcover
Series: 
Wiley Series in Probability and Statistics
Price: 
135.00
ISBN: 
9781118634325
Category: 
Textbook
[Reviewed by
William J. Satzer
, on
03/29/2017
]

This book offers a comprehensive overview of time series analysis. The author says that he aimed the book at a broad audience that includes students, practitioners, and scientists from fields ranging from hydrology to sociology to finance. The focus throughout is on methodologies and techniques selected to help the reader develop a working knowledge of practical applications of time series methods. Most of the book deals with time series that arise from stochastic processes observed at discrete times and recorded at regular intervals. Even restricted in this way, the analysis of time series is a vast subject. The author manages to incorporate a huge number of topics and his book verges on the encyclopedic.

Although the introduction suggests that “some basic knowledge of calculus” is required for understanding most methods described here, that substantially understates the necessary prerequisites. More realistically, these would include a working command of probability and statistics, a good bit of matrix analysis, more than a little real analysis, some knowledge of the Fourier transform and signal processing techniques, and facility with manipulating infinite series. A big dose of mathematical maturity is needed as well.

The book begins with a series of examples of times series arising in applications from several different fields. These illustrate the main features that incur in time series, the kinds of data transformations that can be helpful and circumstances that typically require special treatment. The first chapter also introduces some important notions: serial dependence, stationarity, testing for whiteness of spectrum, parametric and non-parametric modeling, and forecasting.

The main subjects of the book are linear processes, state space models, spectral analysis, estimation methods and prediction. The author also has extended treatments of nonstationary processes, seasonality, regression and the question of how one handles missing values and outliers. Time series analysis, more than any field I’m aware of, is almost overwhelmed by its own acronyms. Practically speaking one can get by with just a few, but happily the author has provides a glossary to help sort out terms like FIGARCH, GARCH, APARCH and LSARFIMA.

It is clear almost from the beginning that this is not a book for those new to the field. A linear process is defined right at the beginning of Chapter 2 in terms of a linear filter on the backshift operator, and then an infinite series expression for the filter is provided. This is very general, and perhaps elegant, but it is hardly illuminating for any reader who might be new to the subject. There are certainly clearer and more transparent ways to define a linear process, and one might have hoped that the author had started there. At least a simple example of two might be warranted. Instead, the author moves on immediately to a discussion of stationarity.

This is a book that would likely be of more use to a serious practitioner of time series analysis than anyone coming fresh to the subject. The author includes too many things, moves through them too quickly and at too high a level to make this anything like an introductory book. For example, the author devotes two pages to the topic of wavelets in discussing spectral analysis. Similarly, observability and controllability (important concepts from control theory) get two pages in the chapter on state space models. This might work if the author were reviewing concepts for well-established practitioners, but it clearly won’t work for other readers.


Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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

Dummy View - NOT TO BE DELETED