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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 and Multidimensional 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 Estim ation .................... ..... .... 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 Periodogram and 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 and Wavelets . . . . . . . . . . ... 232 4.10 Lagged Regression Models . ................... ..245 4.11 Signal Extraction and Optimum Filtering . . . . . . . . . .. . 251 4.12 Spectral Analysis of Multidimensional Series . . . . . . . . ... 256 Problems .... . .................. . . . . . . . .... 258 5 Additional Time Domain Topics 271 5.1 Introduction ........... .... . ............ 271 5.2 Long Memory ARMA and Fractional Differencing . ...... ..271 5.3 GARCH Models . . ........................ ..280 5.4 Threshold Models . .................. . ..... . .. 289 5.5 Regression with Autocorrelated Errors . . . . . . . . . ..... 293 5.6 Lagged Regression: Transfer Function Modeling . . . . . . ... 295 5.7 Multivariate ARMAX Models ... . . . . . . . . . . ..... 302 Problems . . ................. . . . . . . . . . .. 320 6 State-Space Models 324 6.1 Introduction . ......... ................ . ... 324 6.2 Filtering, Smoothing, and Forecasting . . . . . . . . . . .... 330 6.3 Maximum Likelihood Estimation ..... . . . . . . . . . ..... 339 6.4 Missing Data Modifications . . . . . . . . . . . . . . ..348 6.5 Structural Models: Signal Extraction and Forecasting . . . . . 352 6.6 ARMAX Models in State-Space Form . . . . . . . . . ..... 355 6.7 Bootstrapping State-Space Models . . . . . . . . . . ..... 357 6.8 Dynamic Linear Models with Switching . . . . . . . . . .... 362 6.9 Nonlinear and Non-normal State-Space Models Using Monte Carlo Methods . . . . . . . . . . . .... 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 Spectral Matrices 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