Long-Memory Time Series: Theory and Methods

Wilfredo Palma9780470114025, 0470114029

During the last decades long-memory processes have evolved as a vital and important part of time series analysis. This book attempts to give an overview of the theory and methods developed to deal with long-range dependent data as well as describe some applications of these methodologies to real-life time series. The topics are systematically organized in a progressive manner, starting from foundations (the first three chapters), progressing to the analysis of methodological implications (the next six chapters), and finally extending to more complex long-range dependent data structures (the final three chapters).

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Table of contents :
LONG-MEMORY TIME SERIES……Page 4
CONTENTS……Page 8
Preface……Page 16
Acronyms……Page 20
1 Stationary Processes……Page 22
1.1 Fundamental Concepts……Page 23
1.1.1 Stationarity……Page 25
1.1.3 Wold Decomposition Theorem……Page 26
1.1.5 Invertibility……Page 28
1.1.7 Szegö-Kolmogorov Formula……Page 29
1.1.8 Ergodicity……Page 30
1.1.9 Martingales……Page 32
1.1.11 Fractional Brownian Motion……Page 33
1.1.12 Wavelets……Page 35
1.2 Bibliographic Notes……Page 36
Problems……Page 37
2 State Space Systems……Page 42
2.1.2 Hankel Operator……Page 43
2.1.4 Controllability……Page 44
2.2.1 State Space Form to Wold Decomposition……Page 45
2.2.3 Hankel Operator to State Space Form……Page 46
2.3 Estimation of the State……Page 47
2.3.3 State Smoother……Page 48
2.3.5 Steady State System……Page 49
2.3.6 Prediction of Future Observations……Page 51
2.5 Bibliographic Notes……Page 53
Problems……Page 54
3 Long-Memory Processes……Page 60
3.1 Defining Long Memory……Page 61
3.1.1 Alternative Definitions……Page 62
3.2 ARFIMA Processes……Page 64
3.2.1 Stationarity, Causality, and Invertibility……Page 65
3.2.2 Infinite AR and MA Expansions……Page 67
3.2.4 Autocovariance Function……Page 68
3.2.5 Sample Mean……Page 69
3.2.7 lllustrations……Page 70
3.2.8 Approximation of Long-Memory Processes……Page 76
3.3.1 Sample Mean……Page 77
3.4 Technical Lemmas……Page 78
3.5 Bibliographic Notes……Page 79
Problems……Page 80
4 Estimation Methods……Page 86
4.1.2 Durbin-Levinson Algorithm……Page 87
4.1.3 Computation of Autocovariances……Page 88
4.1.4 State Space Approach……Page 90
4.2 Autoregressive Approximations……Page 92
4.2.1 Haslett-Raftery Method……Page 93
4.2.2 Beran Approach……Page 94
4.2.3 A State Space Method……Page 95
4.3 Moving-Average Approximations……Page 96
4.4 Whittle Estimation……Page 99
4.4.2 Non-Gaussian Data……Page 101
4.5 Other Methods……Page 102
4.5.1 A Regression Method……Page 103
4.5.2 Rescaled Range Method……Page 104
4.5.3 Variance Plots……Page 106
4.5.4 Detrended Fluctuation Analysis……Page 108
4.5.5 A Wavelet-Based Method……Page 112
4.6 Numerical Experiments……Page 113
4.7 Bibliographic Notes……Page 114
Problems……Page 115
5 Asymptotic Theory……Page 118
5.1 Notation and Definitions……Page 119
5.2.1 Consistency……Page 120
5.2.2 Central Limit Theorem……Page 122
5.3 Examples……Page 125
5.4 Illustration……Page 129
Problems……Page 130
6 Heteroskedastic Models……Page 136
6.1 Introduction……Page 137
6.2 ARFIMA-GARCH Model……Page 138
6.3 Other Models……Page 140
6.4 Stochastic Volatility……Page 142
6.5 Numerical Experiments……Page 143
6.6.1 Model without Leverage……Page 144
6.6.3 Model Comparison……Page 145
6.7 Bibliographic Notes……Page 146
Problems……Page 147
7 Transformations……Page 152
7.1 Transformations of Gaussian Processes……Page 153
7.2 Autocorrelation of Squares……Page 155
7.3 Asymptotic Behavior……Page 157
7.4 Illustrations……Page 159
7.5 Bibliographic Notes……Page 163
Problems……Page 164
8 Bayesian Methods……Page 168
8.1 Bayesian Modeling……Page 169
8.2.1 Metropolis-Hastings Algorithm……Page 170
8.2.2 Gibbs Sampler……Page 171
8.2.3 Overdispersed Distributions……Page 173
8.3 Monitoring Convergence……Page 174
8.4 A Simulated Example……Page 176
8.5 Data Application……Page 179
Problems……Page 183
9 Prediction……Page 188
9.1.2 Finite Past……Page 189
9.1.3 An Approximate Predictor……Page 193
9.2.1 Infinite Past……Page 194
9.2.2 Finite Past……Page 195
9.3 Heteroskedastic Models……Page 196
9.3.1 Prediction of Volatility……Page 197
9.4 Illustration……Page 199
9.5 Rational Approximations……Page 201
9.5.1 Illustration……Page 203
Problems……Page 205
10 Regression……Page 208
10.1.1 Grenander Conditions……Page 209
10.2 Properties of the LSE……Page 212
10.2.1 Consistency……Page 213
10.2.3 Asymptotic Normality……Page 214
10.3 Properties of the BLUE……Page 215
10.3.1 Efficiency of the LSE Relative to the BLUE……Page 216
10.4.1 Consistency……Page 219
10.4.2 Asymptotic Variance……Page 220
10.4.4 Relative Efficiency……Page 221
10.5 Polynomial Trend……Page 223
10.5.2 Asymptotic Variance……Page 224
10.5.4 Relative Efficiency……Page 225
10.6.3 Normality……Page 226
10.6.4 Efficiency……Page 227
10.7 Illustration: Air Pollution Data……Page 228
10.8 Bibliographic Notes……Page 231
Problems……Page 232
11 Missing Data……Page 236
11.1 Motivation……Page 237
11.2.1 Integration……Page 238
11.2.2 Maximization……Page 239
11.2.4 Kalman Filter with Missing Observations……Page 240
11.3 Effects of Missing Values on ML Estimates……Page 242
11.3.1 Monte Carlo Experiments……Page 243
11.4 Effects of Missing Values on Prediction……Page 244
11.5 Illustrations……Page 248
11.6 Interpolation of Missing Data……Page 250
11.6.1 Bayesian Imputation……Page 255
11.6.2 A Simulated Example……Page 256
Problems……Page 260
12 Seasonality……Page 266
12.1 A Long-Memory Seasonal Model……Page 267
12.2 Calculation of the Asymptotic Variance……Page 271
12.3 Autocovariance Function……Page 273
12.4 Monte Carlo Studies……Page 275
12.5 Illustration……Page 279
12.6 Bibliographic Notes……Page 281
Problems……Page 282
References……Page 286
Topic Index……Page 300
Author Index……Page 304