目录
- Contents
Preface v
1 Introduction 1
1.1 Examples of Time Series 1
1.2 Objectives of Time Series Analysis 9
1.3 Linear Time Series Models 10
1.3.1 White Noise Processes 10
1.3.2 AR Models 10
1.3.3 MA Models 12
1.3.4 ARMA Models 12
1.3.5 AR1MA Models 13
1.4 What 1s a Nonlinear Time Series? 14
1.5 Nonlinear Time Series Models 16
1.5.1 A Simple Example 16
1.5.2 ARCH Models 17
1.5.3 Threshold Models 18
1.5.4 Nonparametric Autoregressive Models 18
1.6 From Linear to Nonlinear Models 20
1.6.1 Local Linear Modeling 20
1.6.2 Global Spline Approximation 23
1.6.3 Goodness-oιFit Tests 24
1 7 Further Reading 25
1.8 Software 1mplementations 27
2 Characteristics of Time Series 29
2.1 Stationarity 29
2 1.1 Definition 29
2 1.2 Stationary ARMA Processes 30
2 1.3 Stationary Gaussian Processes 32
2 1.4 Ergodic Nonlinear Models* 33
2 1.5 Stationary ARCH Processes 37
2.2 Autocorrelation 38
2.2.1 Autocovariance and Autocorrelation 39
2.2.2 Estimation of ACVF and ACF 41
2.2.3 Partial Autocorrelation 43
2.2.4 ACF Plots, PACF Plots, and Examples 45
2.3 Spectral Distributions 48
2.3.1 Periodic Processes 49
2.3.2 Spectral Densities 51
2.3.3 Linear Filters 55
2.4 Periodogram 60
2.4.1 Discrete Fourier 'fransforms 60
2.4.2 Periodogram 62
2.5 Long-Memory Processes* 64
2.5.1 Fractionally Integrated Noise 65
2.5.2 Fractionally Integrated ARMA processes 66
2.6 Mixing* 67
2.6.1 Mixing Conditions 68
2.6.2 Inequalities. 71
2.6.3 Limit Theorems for α-Mixing Processes 74
2.6.4 A Central Limit Theorem for Nonparametric Regression 76
2.7 Complements 78
2.7.1 Proof of Theorem 2.5(i) 78
2.7.2 Proof of Proposition 2.3(i) 79
2.7.3 Proof of Theorem 2.9 79
2.7.4 Proof of Theorem 2.10 80
2.7.5 Proof of Theorem 2.13 81
2.7.6 Proof of Theorem 2.14 81
2.7.7 Proof of Theorem 2.22 84
2.8 Additional Bibliographical Notes 87
3 ARMA Modeling and Forecasting 89
3.1 Models and Background 89
3.2 The Best Linear Prediction一Prewhitening 91
3.3 Maximum Likelihood Estimation 93
3.3.1 Estimators 93
3.3.2 Asymptotic Properties 97
3.3.3 Confidence Intervals 99
3.4 Order Determination 99
3 .4 .1 Akaike lnformation Criterion 100
3 .4 .2 FPE Criterion for AR Modeling 102
3 .4.3 Bayesian lnformation Criterion 103
3 .4.4 Model ldentification 104
3.5 Diagnostic Checking 110
3.5.1 Standardized Residuals 110
3.5.2 Visual Diagnostic. 110
3.5.3 Tests for Whiteness 111
3.6 A Real Data Example-Analyzing German Egg Prices 113
3.7 Linear Forecasting 117
3.7.1 The Least Squares Predictors 117
3.7.2 Forecasting in AR Processes 118
3.7.3 Mean Squared Predictive Errors for AR Processes 119
3.7 .4 Forecasting in ARMA Processes 120
4 Parametric Nonlinear Time Series Models 125
4.1 Threshold Models 125
4 1 1 Threshold Autoregressive Models 126
4 1 2 Estimation and Model ldentification 131
4 1.3 Tests for Linearity 134
4 1.4 Case Studies with Canadian Lynx Data 136
4.2 ARCH and GARCH Models 143
4.2.1 Basic Properties of ARCH Processes 143
4.2.2 Basic Properties of GARCH Processes 147
4.2.3 Estimation 156
4.2 .4 Asymptotic Properties of Conditional MLEs* 161
4.2.5 Bootstrap Confidence lntervals 163
4.2.6 Testing for the ARCH Effect 165
4.2.7 ARCH Modeling of Financial Data 168
4.2.8 A Numerical Example: Modeling S&P 500 lndex Returns 171
4.2.9 Stochastic Volatility Models 179
4.3 Bilinear Models 181
4.3.1 A Simple Example 182
4.3.2 Markovian Rβpresentation 184
4.3.3 Probabilistic Properties* 185
4.3.4 Maximum Likelihood Estimation 189
4.3.5 Bispectrum. 189
4.4 Additional Bibliographical notes 191
5 Nonparametric Density Estimation 193
5.1 lntroduction 193
5.2 Kernel Density Estimation 194
5.3 Windowing and Whitening 197
5.4 Bandwidth Selection 199
5.5 Boundary Correction 202
5.6 Asymptotic Results 204
5.7 Complements-Proof of Theorem 5.3 211
5.8 Bibliographical Notes 212
6 Smoothing in Time Series 215
6.1 Introduction 215
6.2 Smoothing in the Time Domain 215
6.2.1 Trend and Seasonal Components 215
6.2.2 Moving Averages 217
6.2.3 Kernel Smoothing 218
6.2.4 Variations of Kernel Smoothers 220
6.2.5 Filtering 221
6.2.6 Local Linear Smoothing 222
6.2.7 Other Smoothing Methods 224
6.2.8 Seasonal Adjustments 224
6.2.9 Theoretical Aspect♂ 225
6.3 Smoothing in the State Domain 228
6.3.1 Nonparametric Autoregression 228
6.3.2 Local Polynomial Fitting 230
6.3.3 Properties of the Local Polynomial Estimator 234
6.3.4 Standard Errors and Estimated Bias 241
6.3.5 Bandwidth Selection 243
6.4 Spline Methods 246
6.4.1 Polynomial Splines 247
6.4.2 Nonquadratic Penalized Splines 249
6.4.3 Smoothing Splines 251
6.5 Estimation of Conditional Densities 253
6.5.1 Methods of Estimation 253
6.5.2 Asymptotic Properties* 256
6.6 Complements 257
6.6.1 Proof of Theorem 6.1 257
6.6.2 Conditions and Proof of Theorem 6.3 260
6.6.3 Proof of Lemma 6.1 266
6.6.4 Proof of Theorem 6.5 268
6.6.5 Proof for Theorems 6.6 and 6.7 269
6.7 Bibliographical Notes 271
7 Spectral Density Estimation and Its Applications 275
7.1 Introduction 275
7.2 Tapering, Kernel Estimation, and Prewhitening 276
7.2.1 Tapering 277
7.2.2 Smoothing the Periodogram 281
7.2.3 Prewhitening and Bias Reduction 282
7.3 Automatic Estimation of Spectral Density 283
7.3.1 Least-Squares Estimators and Bandwidth Selection 284
7.3.2 Local Maximum Likelihood Estimator 286
7.3.3 Confidence Intervals 289
7.4 Tests for White Noise 296
7.4.1 Fisher's Test 296
7.4.2 Generalized Likelihood Ratio Test 298
7 .4.3 泸-Test and the Adaptive Neyman Test 300
7ι4 Ot her Smoothing-Based Tests 302
7.4.5 Numerical Examples 303
7.5 Complements 304
7.5.1 Conditions for Theorems 7.1一-7.3 304
7.5.2 Lemmas 305
7.5.3 Proof of Theorem 7.1 306
7.5.4 Proof of Theorem 7.2 307
7.5.5 Proof of Theorem 7.3 307
7.6 Bibliographical Notes 310
8 Nonparametric Models 313
8.1 Introduction . 313
8.2 Multivariate Local Polynomial Regression 314
8.2.1 Multivariate Kernel Functions 314
8.2.2 Multivariate Local Linear Regression 316
8.2.3 Mult ivariate Local Quadratic Regression 317
8.3 Functional-Coefficient Autoregressive Model 318
8.3.1 The Model 318
8.3.2 Relation to Stochastic Regression 318
8.3.3 Ergodicity* 319
8.3.4 Estimat ion of Coefficient Functions 321
8.3.5 Selection of Bandwidth and Model-Dependent Variable322
8.3.6 Prediction . 324
8.3.7 Examples 324
8.3.8 Sampling Properties* . 332
8.4 Adaptive Functional-Coefficient Autoregressive Models 333
8.4.1 The Models 334
8.4.2 Existence and Identifiability 335
8.4.3 Profile Least-Squares Estimation 337
8.4.4 Bandwidth Selection 340
8.4.5 Variable Selection 340
8.4.6 Implementation 341
8.4.7 Examples 343
8.4.8 Extensions 349
8.5 Additive Models 349
8.5.1 The Models 349
8.5.2 The Backfitting Algorithm 350
8.5.3 Projections and A verage Surface Estimators 352
8.5.4 Estimability of Coefficient Functions 354
8.5.5 Bandwidth Selection 355
8.5.6 Examples 356
8.6 Other Nonparametric Models 364
8.6.1 Two-Term Interaction Models 365
8.6.2 Partially Linear Models 366
8.6.3 Single-Index Models 367
8.6.4 Multiple-Index Models 368
8.6.5 An Analysis of Environmental Data 371
8.7 Modeling Conditional Variance 374
8.7.1 Methods of Estimating Conditional Variance 375
8.7.2 Univariate Setting 376
8.7.3 Functional-Coefficient Models 382
8.7.4 Additive Models 382
8.7.5 Product Models 384
8.7.6 Other Nonparametric Models 384
8.8 Complements 384
8.8.1 Proof of Theorem 8.1 384
8.8.2 Technical Conditions for Theorems 8.2 and 8.3 386
8.8.3 Preliminaries to the Proof of Theorem 8.3 387
8.8.4 Proof of Theorem 8.3 390
8.8.5 Proof of Theorem 8.4 392
8.8.6 Conditions of Theorem 8.5 394
8.8.7 Proof of Theorem 8.5 395
8.9 Bibliographical Notes 399
9 Model Validation 405
9.1 Introduction 405
9.2 Generalized Likelihood Ratio Tests 406
9.2.1 Introduction 406
9.2.2 Generalized Likelihood Ratio Test 408
9.2.3 Null Distributions and the Bootstrap 409
9.2.4 Power of the GLR Test 414
9.2.5 Bias Reduction 414
9.2.6 Nonparametric versus Nonparametric Models 415
9.2.7 Choice of Bandwidth 416
9.2.8 A Numerical Example 417
9.3 Tests on Spectral Densities 419
9.3.1 Relation with Nonparametric Regression 421
9.3.2 Generalized Likelihood Ratio Tests 421
9.3.3 Other Nonparametric Methods 425
9.3.4 Tests Based on Rescaled Periodogram 427
9.4 Autoregressive versus Nonparametric Models 430
9.4.1 Functional-Coefficient Alternatives 430
9.4.2 Additive Alternatives 434
9.5 Threshold Models versus Varying-Coefficient Models 437
9.6 Bibliographical Notes 439
10 Nonlinear Prediction 441
10.1 Features of Nonlinear Prediction 441
10.1.1 Decomposition for Mean Square Predictive Errors 441
10.1.2 Noise Amplification 444
10 1.3 Sensitivity to Initial Values 445
10 1.4 Multiple-Step Prediction versus a One-Step Plug-inMethod 447
10 1.5 Nonlinear versus Linear Prediction 448
10.2 Point Prediction 450
10.2.1 Local Linear Predictors 450
10.2.2 An Example 451
10.3 Estimating Predictive Distributions 454
10.3.1 Local Logistic Estimator 455
10.3.2 Adjusted Nadaraya-Watson Estimator 456
10.3.3 Bootstrap Bandwidth Selection 457
10.3.4 Numerical Examples 458
10.3.5 Asymptotic Properties 463
10.3.6 Sensitivity to Initial Values: A Conditional DistributionApproach 466
10.4 Interval Predictors and Predictive Sets 470
10.4.1 Minimum-Length Predictive Sets 471
10.4.2 Estimation of Minimum-Length Predictors 474
10.4.3 Numerical Examples 476
10.5 Complements 482
10.6 Additional Bibliographical Notes 485
References 487
Author index 537
Subject index 545
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