Mathematical Foundations of Time Series Analysis (eBook)

Jan Beran is Professor of Statistics at the Department of Mathematics and Statistics at the University of Konstanz, Germany. After completing his Ph.D. in mathematics at the ETH Zurich, Switzerland, he worked at several universities in the USA and at the University of Zurich in Switzerland. He has a broad range of interests, from long-memory processes and asymptotic theory to applications in finance, biology, and musicology.

Preface 5
Contents 6
1 Introduction 9
1.1 What Is a Time Series? 9
1.2 Time Series Versus iid Data 10
2 Typical Assumptions 13
2.1 Fundamental Properties 13
2.1.1 Ergodic Property with a Constant Limit 13
2.1.2 Strict Stationarity 15
2.1.3 Weak Stationarity 16
2.1.4 Weak Stationarity and Hilbert Spaces 19
2.1.5 Ergodic Processes 40
2.1.6 Sufficient Conditions for the a.s. Ergodic Property with a Constant Limit 42
2.1.7 Sufficient Conditions for the L2-Ergodic Property with a Constant Limit 43
2.2 Specific Assumptions 47
2.2.1 Gaussian Processes 47
2.2.2 Linear Processes in L2(?) 48
2.2.3 Linear Processes with E(Xt2)=? 52
2.2.4 Multivariate Linear Processes 56
2.2.5 Invertibility 57
2.2.6 Restrictions on the Dependence Structure 71
2.2.6.1 Markov Processes 71
2.2.6.2 Mixing Conditions 76
2.2.6.3 Short Memory, Long Memory, Antipersistence 76
3 Defining Probability Measures for Time Series 77
3.1 Finite Dimensional Distributions 77
3.2 Transformations and Equations 78
3.3 Conditions on the Expected Value 79
3.4 Conditions on the Autocovariance Function 81
3.4.1 Positive Semidefinite Functions 81
3.4.2 Spectral Distribution 85
3.4.3 Calculation and Properties of F and f 94
4 Spectral Representation of Univariate Time Series 109
4.1 Motivation 109
4.2 Harmonic Processes 110
4.3 Extension to General Processes 113
4.3.1 Stochastic Integrals with Respect to Z 113
4.3.2 Existence and Definition of Z 120
4.3.2.1 Existence 120
4.3.2.2 Calculation of Z 128
4.3.3 Interpretation of the Spectral Representation 130
4.4 Further Properties 130
4.4.1 Relationship Between Re Z and Im Z 130
4.4.2 Frequency 131
4.4.3 Overtones 132
4.4.4 Why Are Frequencies Restricted to the Range [-?,?]? 133
4.5 Linear Filters and the Spectral Representation 137
4.5.1 Effect on the Spectral Representation 137
4.5.2 Elimination of Frequency Bands 142
5 Spectral Representation of Real Valued Vector Time Series 144
5.1 Cross-Spectrum and Spectral Representation 144
5.2 Coherence and Phase 153
6 Univariate ARMA Processes 167
6.1 Definition 167
6.2 Stationary Solution 167
6.3 Causal Stationary Solution 172
6.4 Causal Invertible Stationary Solution 175
6.5 Autocovariances of ARMA Processes 176
6.5.1 Calculation by Integration 176
6.5.2 Calculation Using the Autocovariance Generating Function 176
6.5.3 Calculation Using the Wold Representation 181
6.5.4 Recursive Calculation 182
6.5.5 Asymptotic Decay 183
6.6 Integrated, Seasonal and Fractional ARMA and ARIMA Processes 191
6.6.1 Integrated Processes 191
6.6.2 Seasonal ARMA Processes 192
6.6.3 Fractional ARIMA Processes 193
6.7 Unit Roots, Spurious Correlation, Cointegration 206
7 Generalized Autoregressive Processes 209
7.1 Definition of Generalized Autoregressive Processes 209
7.2 Stationary Solution of Generalized Autoregressive Equations 210
7.3 Definition of VARMA Processes 215
7.4 Stationary Solution of VARMA Equations 217
7.5 Definition of GARCH Processes 219
7.6 Stationary Solution of GARCH Equations 220
7.7 Definition of ARCH(?) Processes 225
7.8 Stationary Solution of ARCH(?) Equations 226
8 Prediction 229
8.1 Best Linear Prediction Given an Infinite Past 229
8.2 Predictability 231
8.3 Construction of the Wold Decomposition from f 236
8.4 Best Linear Prediction Given a Finite Past 241
9 Inference for ?, ? and F 246
9.1 Location Estimation 246
9.2 Linear Regression 249
9.3 Nonparametric Estimation of ? 258
9.4 Nonparametric Estimation of f 267
10 Parametric Estimation 286
10.1 Gaussian and Quasi Maximum Likelihood Estimation 286
10.2 Whittle Approximation 289
10.3 Autoregressive Approximation 292
10.4 Model Choice 294
References 297
Author Index 302
Subject Index 305