Time Series Analysis and Its Applications: With R Examples

Contents
1 Characteristics of Time Series 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Nature of Time Series Data . . . . . . . . . . . . . . . . . 4
1.3 Time Series Statistical Models . . . . . . . . . . . . . . . . . . . 11
1.4 Measures of Dependence: Autocorrelation
and Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Stationary Time Series . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Estimation of Correlation . . . . . . . . . . . . . . . . . . . . . 29
1.7 Vector-Valued andMultidimensional Series . . . . . . . . . . . 34
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Time Series Regression and Exploratory Data Analysis 48
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 Classical Regression in the Time Series Context . . . . . . . . . 49
2.3 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . 57
2.4 Smoothing in the Time Series Context . . . . . . . . . . . . . . 71
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3 ARIMA Models 84
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.2 Autoregressive Moving Average Models . . . . . . . . . . . . . 85
3.3 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 98
3.4 Autocorrelation and Partial Autocorrelation Functions . . . . . 103
3.5 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.7 Integrated Models for Nonstationary Data . . . . . . . . . . . . 140
3.8 Building ARIMA Models . . . . . . . . . . . . . . . . . . . . . 143
3.9 Multiplicative Seasonal ARIMA Models . . . . . . . . . . . . . 154
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4 Spectral Analysis and Filtering 174
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.2 Cyclical Behavior and Periodicity . . . . . . . . . . . . . . . . . 176
4.3 The Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 181
4.4 Periodogramand Discrete Fourier Transform . . . . . . . . . . 187
4.5 Nonparametric Spectral Estimation . . . . . . . . . . . . . . . . 197
4.6 Multiple Series and Cross-Spectra . . . . . . . . . . . . . . . . . 215
4.7 Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.8 Parametric Spectral Estimation . . . . . . . . . . . . . . . . . . 228
4.9 Dynamic Fourier Analysis andWavelets . . . . . . . . . . . . . 232
4.10 Lagged Regression Models . . . . . . . . . . . . . . . . . . . . 245
4.11 Signal Extraction and Optimum Filtering . . . . . . . . . . . . 251
4.12 Spectral Analysis ofMultidimensional Series . . . . . . . . . . . 256
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5 Additional Time Domain Topics 271
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
5.2 LongMemory ARMA and Fractional Differencing . . . . . . . 271
5.3 GARCHModels . . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.4 ThresholdModels . . . . . . . . . . . . . . . . . . . . . . . . . . 289
5.5 Regression with Autocorrelated Errors . . . . . . . . . . . . . . 293
5.6 Lagged Regression: Transfer Function Modeling . . . . . . . . . 295
5.7 Multivariate ARMAXModels . . . . . . . . . . . . . . . . . . . 302
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
6 State-Space Models 324
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
6.2 Filtering, Smoothing, and Forecasting . . . . . . . . . . . . . . 330
6.3 MaximumLikelihood Estimation . . . . . . . . . . . . . . . . . 339
6.4 Missing Data Modifications . . . . . . . . . . . . . . . . . . . . 348
6.5 StructuralModels: Signal Extraction and Forecasting . . . . . 352
6.6 ARMAX Models in State-Space Form . . . . . . . . . . . . . . 355
6.7 Bootstrapping State-Space Models . . . . . . . . . . . . . . . . 357
6.8 Dynamic LinearModels with Switching . . . . . . . . . . . . . 362
6.9 Nonlinear and Non-normal State-Space
Models UsingMonte CarloMethods . . . . . . . . . . . . . . . 376
6.10 Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . 388
6.11 State-Space and ARMAX Models for
Longitudinal Data Analysis . . . . . . . . . . . . . . . . . . . . 394
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
7 Statistical Methods in the Frequency Domain 412
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
7.2 SpectralMatrices and Likelihood Functions . . . . . . . . . . . 416
7.3 Regression for Jointly Stationary Series . . . . . . . . . . . . . 417
7.4 Regression with Deterministic Inputs . . . . . . . . . . . . . . . 426
7.5 Random Coefficient Regression . . . . . . . . . . . . . . . . . . 434
7.6 Analysis of Designed Experiments . . . . . . . . . . . . . . . . 438
7.7 Discrimination and Cluster Analysis . . . . . . . . . . . . . . . 449
7.8 Principal Components and Factor Analysis . . . . . . . . . . . 464
7.9 The Spectral Envelope . . . . . . . . . . . . . . . . . . . . . . . 479
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
Appendix A: Large Sample Theory 501
A.1 ConvergenceModes . . . . . . . . . . . . . . . . . . . . . . . . 501
A.2 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . 509
A.3 TheMean and Autocorrelation Functions . . . . . . . . . . . . 513
Appendix B: Time Domain Theory 522
B.1 Hilbert Spaces and the Projection Theorem . . . . . . . . . . . 522
B.2 Causal Conditions for ARMAModels . . . . . . . . . . . . . . 526
B.3 Large Sample Distribution of the AR(p)
Conditional Least Squares Estimators . . . . . . . . . . . . . . 528
B.4 TheWold Decomposition . . . . . . . . . . . . . . . . . . . . . 532
Appendix C: Spectral Domain Theory 534
C.1 Spectral Representation Theorem . . . . . . . . . . . . . . . . . 534
C.2 Large Sample Distribution of the DFT and
Smoothed Periodogram . . . . . . . . . . . . . . . . . . . . . . 539
C.3 The ComplexMultivariate Normal Distribution . . . . . . . . . 550
References 555
Index 569

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