# Time Series Analysis with Long Memory in View

## Books

Provides a simple exposition of the basic time series material, and insights into underlying technical aspects and methods of proof

Long memory time series are characterized by a strong dependence between distant events. This book introduces readers to the theory and foundations of univariate time series analysis with a focus on long memory and fractional integration, which are embedded into the general framework. It presents the general theory of time series, including some issues that are not treated in other books on time series, such as ergodicity, persistence versus memory, asymptotic properties of the periodogram, and Whittle estimation.  Further chapters address the general functional central limit theory, parametric and semiparametric estimation of the long memory parameter, and locally optimal tests.

Intuitive and easy to read, Time Series Analysis with Long Memory in View offers chapters that cover: Stationary Processes; Moving Averages and Linear Processes; Frequency Domain Analysis; Differencing and Integration; Fractionally Integrated Processes; Sample Means; Parametric Estimators; Semiparametric Estimators; and Testing. It also discusses further topics. This book:

• Offers beginning-of-chapter examples as well as end-of-chapter technical arguments and proofs
• Contains many new results on long memory processes which have not appeared in previous and existing textbooks
• Takes a basic mathematics (Calculus) approach to the topic of time series analysis with long memory
• Contains 25 illustrative figures as well as lists of notations and acronyms

Time Series Analysis with Long Memory in View is an ideal text for first year PhD students, researchers, and practitioners in statistics, econometrics, and any application area that uses time series over a long period. It would also benefit researchers, undergraduates, and practitioners in those areas who require a rigorous introduction to time series analysis.

List of Figures xi

Preface xiii

List of Notation xv

Acronyms xvii

1 Introduction 1

1.1 Empirical Examples 1

1.2 Overview 6

2 Stationary Processes 11

2.1 Stochastic Processes 11

2.2 Ergodicity 14

2.3 Memory and Persistence 22

2.4 Technical Appendix: Proofs 25

3 Moving Averages and Linear Processes 27

3.1 Infinite Series and Summability 27

3.2 Wold Decomposition and Invertibility 32

3.3 Persistence versus Memory 37

3.4 Autoregressive Moving Average Processes 47

3.5 Technical Appendix: Proofs 51

4 Frequency Domain Analysis 57

4.1 Decomposition into Cycles 57

4.2 Complex Numbers and Transfer Functions 62

4.3 The Spectrum 63

4.4 Parametric Spectra 68

4.5 (Asymptotic) Properties of the Periodogram 72

4.6 Whittle Estimation 76

4.7 Technical Appendix: Proofs 81

5 Differencing and Integration 89

5.1 Integer Case 89

5.2 Approximating Sequences and Functions 91

5.3 Fractional Case 95

5.4 Technical Appendix: Proofs 99

6 Fractionally Integrated Processes 103

6.1 Definition and Properties 103

6.2 Examples and Discussion 108

6.3 Nonstationarity and Type I Versus II 114

6.4 Practical Issues 118

6.5 Frequency Domain Assumptions 120

6.6 Technical Appendix: Proofs 123

7 Sample Mean 127

7.1 Central Limit Theorem for I(0) Processes 127

7.2 Central Limit Theorem for I(d) Processes 129

7.3 Functional Central Limit Theory 132

7.4 Inference About the Mean 139

7.5 Sample Autocorrelation 141

7.6 Technical Appendix: Proofs 145

8 Parametric Estimators 149

8.1 Parametric Assumptions 149

8.2 Exact Maximum Likelihood Estimation 150

8.3 Conditional Sum of Squares 154

8.4 Parametric Whittle Estimation 156

8.5 Log-periodogram Regression of FEXP Processes 161

8.6 Fractionally Integrated Noise 164

8.7 Technical Appendix: Proofs 165

9 Semiparametric Estimators 169

9.1 Local Log-periodogram Regression 169

9.2 Local Whittle Estimation 175

9.3 Finite Sample Approximation 182

9.4 Bias Approximation and Reduction 184

9.5 Bandwidth Selection 188

9.6 Global Estimators 193

9.7 Technical Appendix: Proofs 195

10 Testing 197

10.1 Hypotheses on Fractional Integration 197

10.2 Rescaled Range or Variance 199

10.3 The Score Test Principle 204

10.4 Lagrange Multiplier (LM) Test 205

10.5 LM Test in the Frequency Domain 210

10.6 Regression-based LM Test 213

10.7 Technical Appendix: Proofs 218

11 Further Topics 223

11.1 Model Selection and Specification Testing 223

11.2 Spurious Long Memory 226

11.3 Forecasting 229

11.4 Cyclical and Seasonal Models 231

11.5 Long Memory in Volatility 234

11.6 Fractional Cointegration 236

11.7 R Packages 240

11.8 Neglected Topics 241

Bibliography 245

Index 267

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