CHAPTER 1 Stationary Time Series 1
1.1 Examples of Time Series 1
1.2 Stochastic Processes 8
1.3 Stationarity and Strict Stationarity 11
1.4 The Estimation and Elimination of Trend and Seasonal Components 14
1.5 The Autocovariance Function of a Stationary Process 25
1.6 The Multivariate Normal Distribution 32
1.7 Applications of Kolmogorov's Theorem 37
Problems 39
CHAPTER 2 Hilbert Spaces 42
2.1 Inner-Product Spaces and Their Properties 42
2.2 Hilbert Spaces 46
2.3 The Projection Theorem 48
2.4 Orthonormal Sets 54
2.5 Projection in Rn 58
2.6 Linear Regression and the General Linear Model 60
2.7 Mean Square Convergence,Conditional Expectation and Best Linear Prediction in L2(？,？,P) 62
2.8 Fourier Series 65
2.9 Hilbert Space Isomorphisms 67
2.10 The Completeness of L2(？,？,P) 68
2.11 Complementary Results for Fourier Series 69
Problems 73
CHAPTER 3 Stationary ARMA Processes 77
3.1 Causal and Invertible ARMA Processes 77
3.2 Moving Average Processes of Infinite Order 89
3.3 Computing the Autocovariance Function of an ARMA(p,q)Process 91
3.4 The Partial Autocorrelation Function 98
3.5 The Autocovariance Generating Function 103
3.6 Homogeneous Linear Difference Equations with Constant Coefficients 105
Problems 110
CHAPTER 4 The Spectral Representation of a Stationary Process 114
4.1 Complex-Valued Stationary Time Series 114
4.2 The Spectral Distribution of a Linear Combination of Sinusoids 116
4.3 Herglotz's Theorem 117
4.4 Spectral Densities and ARMA Processes 122
4.5 Circulants and Their Eigenvalues 133
4.6 Orthogonal Increment Processes on[-π,π] 138
4.7 Integration with Respect to an Orthogonal Increment Process 140
4.8 The Spectral Representation 143
4.9 Inversion Formulae 150
4.10 Time-Invariant Linear Filters 152
4.11 Properties of the Fourier Approximation hn to I(ν,ω] 157
Problems 159
CHAPTER 5 Prediction of Stationary Processes 166
5.1 The Prediction Equations in the Time Domain 166
5.2 Recursive Methods for Computing Best Linear Predictors 169
5.3 Recursive Prediction of an ARMA(p,q)Process 175
5.4 Prediction of a Stationary Gaussian Process;Prediction Bounds 182
5.5 Prediction of a Causal Invertible ARMA Process in Terms of Xj,-∞<j≤n 182
5.6 Prediction in the Frequency Domain 185
5.7 The Wold Decomposition 187
5.8 Kolmogorov's Formula 191
Problems 192
CHAPTER 6 Asymptotic Theory 198
6.1 Convergence in Probability 198
6.2 Convergence in rth Mean,r>0 202
6.3 Convergence in Distribution 204
6.4 Central Limit Theorems and Related Results 209
Problems 215
CHAPTER 7 Estimation of the Mean and the Autocovariance Function 218
7.1 Estimation of μ 218
7.2 Estimation of γ(·)andρ(·) 220
7.3 Derivation of the Asymptotic Distributions 225
Problems 236
CHAPTER 8 Estimation for ARMA Models 238
8.1 The Yule-Walker Equations and Parameter Estimation for Autoregressive Processes 239
8.2 Preliminary Estimation for Autoregressive Processes Using the Durbin-Levinson Algorithm 241
8.3 Preliminary Estimation for Moving Average Processes Using the Innovations Algorithm 245
8.4 Preliminary Estimation for ARMA(p,q)Processes 250
8.5 Remarks on Asymptotic Efficiency 253
8.6 Recursive Calculation of the Likelihood of an Arbitrary Zero-Mean Gaussian Process 254
8.7 Maximum Likelihood and Least Squares Estimation for ARMA Processes 256
8.8 Asymptotic Properties of the Maximum Likelihood Estimators 258
8.9 Confidence Intervals for the Parameters of a Causal Invertible ARMA Process 260
8.10 Asymptotic Behavior of the Yule-Walker Estimates 262
8.11 Asymptotic Normality of Parameter Estimators 265
Problems 269
CHAPTER 9 Model Building and Forecasting with ARIMA Processes 273
9.1 ARIMA Models for Non-Stationary Time Series 274
9.2 Identification Techniques 284
9.3 Order Selection 301
9.4 Diagnostic Checking 306
9.5 Forecasting ARIMA Models 314
9.6 Seasonal ARIMA Models 320
Problems 326
CHAPTER 10 Inference for the Spectrum of a Stationary Process 330
10.1 The Periodogram 331
10.2 Testing for the Presence of Hidden Periodicities 334
10.3 Asymptotic Properties of the Periodogram 342
10.4 Smoothing the Periodogram 350
10.5 Confidence Intervals for the Spectrum 362
10.6 Autoregressive,Maximum Entropy,Moving Average and Maximum Likelihood ARMA Spectral Estimators 365
10.7 The Fast Fourier Transform(FFT)Algorithm 373
10.8 Derivation of the Asymptotic Behavior of the Maximum Likelihood and Least Squares Estimators of the Coefficients of an ARMA Process 375
Problems 396
CHAPTER 11 Multivariate Time Series 401
11.1 Second Order Properties of Multivariate Time Series 402
11.2 Estimation of the Mean and Covariance Function 405
11.3 Multivariate ARMA Processes 417
11.4 Best Linear Predictors of Second Order Random Vectors 421
11.5 Estimation for Multivariate ARMA Processes 430
11.6 The Cross Spectrum 434
11.7 Estimating the Cross Spectrum 443
11.8 The Spectral Representation of a Multivariate Stationary Time Series 454
Problems 459
CHAPTER 12 State-Space Models and the Kalman Recursions 463
12.1 State-Space Models 463
12.2 The Kalman Recursions 474
12.3 State-Space Models with Missing Observations 482
12.4 Controllability and Observability 489
12.5 Recursive Bayesian State Estimation 498
Problems 501
CHAPTER 13 Further Topics 506
13.1 Transfer Function Modelling 506
13.2 Long Memory Processes 520
13.3 Linear Processes with Infinite Variance 535
13.4 Threshold Models 545
Problems 552
Appendix:Data Sets 555
Bibliography 561
Index 567