Long-Memory Time Series Theory and Methods

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ISBN: 9780470114025 | 0470114029
Cover: Hardcover
Copyright: 3/30/2007

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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).

Preface	p. xiii
Acronyms	p. xvii
Stationary Processes	p. 1
Fundamental Concepts	p. 2
Stationarity	p. 4
Singularity and Regularity	p. 5
Wold Decomposition Theorem	p. 5
Causality	p. 7
Invertibility	p. 7
Best Linear Predictor	p. 8
Szego-Kolmogorov Formula	p. 8
Ergodicity	p. 9
Martingales	p. 11
Cumulants	p. 12
Fractional Brownian Motion	p. 12
Wavelets	p. 14
Bibliographic Notes	p. 15
Problems	p. 16
State Space Systems	p. 21
Introduction	p. 22
Stability	p. 22
Hankel Operator	p. 22
Observability	p. 23
Controllability	p. 23
Minimality	p. 24
Representations of Linear Processes	p. 24
State Space Form to Wold Decomposition	p. 24
Wold Decomposition to State Space Form	p. 25
Hankel Operator to State Space Form	p. 25
Estimation of the State	p. 26
State Predictor	p. 27
State Filter	p. 27
State Smoother	p. 27
Missing Observations	p. 28
Steady State System	p. 28
Prediction of Future Observations	p. 30
Extensions	p. 32
Bibliographic Notes	p. 32
Problems	p. 33
Long-Memory Processes	p. 39
Defining Long Memory	p. 40
Alternative Definitions	p. 41
Extensions	p. 43
ARFIMA Processes	p. 43
Stationarity, Causality, and Invertibility	p. 44
Infinite AR and MA Expansions	p. 46
Spectral Density	p. 47
Autocovariance Function	p. 47
Sample Mean	p. 48
Partial Autocorrelations	p. 49
Illustrations	p. 49
Approximation of Long-Memory Processes	p. 55
Fractional Gaussian Noise	p. 56
Sample Mean	p. 56
Technical Lemmas	p. 57
Bibliographic Notes	p. 58
Problems	p. 59
Estimation Methods	p. 65
Maximum-Likelihood Estimation	p. 66
Cholesky Decomposition Method	p. 66
Durbin-Levinson Algorithm	p. 66
Computation of Autocovariances	p. 67
State Space Approach	p. 69
Autoregressive Approximations	p. 71
Haslett-Raftery Method	p. 72
Beran Approach	p. 73
A State Space Method	p. 74
Moving-Average Approximations	p. 75
Whittle Estimation	p. 78
Other versions	p. 80
Non-Gaussian Data	p. 80
Semiparametric Methods	p. 81
Other Methods	p. 81
A Regression Method	p. 82
Rescaled Range Method	p. 83
Variance Plots	p. 85
Detrended Fluctuation Analysis	p. 87
A Wavelet-Based Method	p. 91
Numerical Experiments	p. 92
Bibliographic Notes	p. 93
Problems	p. 94
Asymptotic Theory	p. 97
Notation and Definitions	p. 98
Theorems	p. 99
Consistency	p. 99
Central Limit Theorem	p. 101
Efficiency	p. 104
Examples	p. 104
Illustration	p. 108
Technical Lemmas	p. 109
Bibliographic Notes	p. 109
Problems	p. 109
Heteroskedastic Models	p. 115
Introduction	p. 116
ARFIMA-GARCH Model	p. 117
Estimation	p. 119
Other Models	p. 119
Estimation	p. 121
Stochastic Volatility	p. 121
Estimation	p. 122
Numerical Experiments	p. 122
Application	p. 123
Model without Leverage	p. 123
Model with Leverage	p. 124
Model Comparison	p. 124
Bibliographic Notes	p. 125
Problems	p. 126
Transformations	p. 131
Transformations of Gaussian Processes	p. 132
Autocorrelation of Squares	p. 134
Asymptotic Behavior	p. 136
Illustrations	p. 138
Bibliographic Notes	p. 142
Problems	p. 143
Bayesian Methods	p. 147
Bayesian Modeling	p. 148
Markov Chain Monte Carlo Methods	p. 149
Metropolis-Hastings Algorithm	p. 149
Gibbs Sampler	p. 150
Overdispersed Distributions	p. 152
Monitoring Convergence	p. 153
A Simulated Example	p. 155
Data Application	p. 158
Bibliographic Notes	p. 162
Problems	p. 162
Prediction	p. 167
One-Step Ahead Predictors	p. 168
Infinite Past	p. 168
Finite Past	p. 168
An Approximate Predictor	p. 172
Multistep Ahead Predictors	p. 173
Infinite Past	p. 173
Finite Past	p. 174
Heteroskedastic Models	p. 175
Prediction of Volatility	p. 176
Illustration	p. 178
Rational Approximations	p. 180
Illustration	p. 182
Bibliographic Notes	p. 184
Problems	p. 184
Regression	p. 187
Linear Regression Model	p. 188
Grenander Conditions	p. 188
Properties of the LSE	p. 191
Consistency	p. 192
Asymptotic Variance	p. 193
Asymptotic Normality	p. 193
Properties of the BLUE	p. 194
Efficiency of the LSE Relative to the BLUE	p. 195
Estimation of the Mean	p. 198
Consistency	p. 198
Asymptotic Variance	p. 199
Normality	p. 200
Relative Efficiency	p. 200
Polynomial Trend	p. 202
Consistency	p. 203
Asymptotic Variance	p. 203
Normality	p. 204
Relative Efficiency	p. 204
Harmonic Regression	p. 205
Consistency	p. 205
Asymptotic Variance	p. 205
Normality	p. 205
Efficiency	p. 206
Illustration: Air Pollution Data	p. 207
Bibliographic Notes	p. 210
Problems	p. 211
Missing Data	p. 215
Motivation	p. 216
Likelihood Function with Incomplete Data	p. 217
Integration	p. 217
Maximization	p. 218
Calculation of the Likelihood Function	p. 219
Kalman Filter with Missing Observations	p. 219
Effects of Missing Values on ML Estimates	p. 221
Monte Carlo Experiments	p. 222
Effects of Missing Values on Prediction	p. 223
Illustrations	p. 227
Interpolation of Missing Data	p. 229
Bayesian Imputation	p. 234
A Simulated Example	p. 235
Bibliographic Notes	p. 239
Problems	p. 239
Seasonality	p. 245
A Long-Memory Seasonal Model	p. 246
Calculation of the Asymptotic Variance	p. 250
Autocovariance Function	p. 252
Monte Carlo Studies	p. 254
Illustration	p. 258
Bibliographic Notes	p. 260
Problems	p. 261
References	p. 265
Topic Index	p. 279
Author Index	p. 283
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Long-Memory Time Series Theory and Methods

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