# Nonlinear Time Series: Nonparametric and Parametric Methods

Springer Science & Business Media, Nov 23, 2007 - Mathematics - 553 pages
Amongmanyexcitingdevelopmentsinstatisticsoverthelasttwodecades, nonlineartimeseriesanddata-analyticnonparametricmethodshavegreatly advanced along seemingly unrelated paths. In spite of the fact that the - plication of nonparametric techniques in time series can be traced back to the 1940s at least, there still exists healthy and justi?ed skepticism about the capability of nonparametric methods in time series analysis. As - thusiastic explorers of the modern nonparametric toolkit, we feel obliged to assemble together in one place the newly developed relevant techniques. Theaimofthisbookistoadvocatethosemodernnonparametrictechniques that have proven useful for analyzing real time series data, and to provoke further research in both methodology and theory for nonparametric time series analysis. Modern computers and the information age bring us opportunities with challenges. Technological inventions have led to the explosion in data c- lection (e.g., daily grocery sales, stock market trading, microarray data). The Internet makes big data warehouses readily accessible. Although cl- sic parametric models, which postulate global structures for underlying systems, are still very useful, large data sets prompt the search for more re?nedstructures,whichleadstobetterunderstandingandapproximations of the real world. Beyond postulated parametric models, there are in?nite other possibilities. Nonparametric techniques provide useful exploratory tools for this venture, including the suggestion of new parametric models and the validation of existing ones.

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### Contents

 Introduction 1 12 Objectives of Time Series Analysis 9 13 Linear Time Series Models 10 133 MA Models 12 135 A RIM A Models 13 14 What Is a Nonlinear Time Series? 14 15 Nonlinear Time Series Models 16 152 ARCH Models 17
 66 Complements 257 662 Conditions and Proof of Theorem 63 260 663 Proof of Lemma 61 266 664 Proof of Theorem 65 268 665 Proof for Theorems 66 and 67 269 67 Bibliographical Notes 271 Spectral Density Estimation and Its Applications 275 72 Tapering Kernel Estimation and Prewhitening 276

 153 Threshold Models 18 16 From Linear to Nonlinear Models 20 162 Global Spline Approximation 23 163 GoodnessofFit Tests 24 17 Further Reading 25 18 Software Implementations 27 Characteristics of Time Series 29 212 Stationary ARMA Processes 30 213 Stationary Gaussian Processes 32 214 Ergodic Nonlinear Models 33 215 Stationary ARCH Processes 37 22 Autocorrelation 38 221 Autocovariance and Autocorrelation 39 222 Estimation oACVF and ACF 41 223 Partial Autocorrelation 43 224 ACF Plots PACF Plots and Examples 45 23 Spectral Distributions 48 231 Periodic Processes 49 232 Spectral Densities 51 233 Linear Filters 55 24 Periodogram 60 242 Periodogram 62 25 LongMemory Processes 64 251 Fractionally Integrated Noise 65 252 Fractionally Integrated ARMA processes 66 26 Mixing 67 261 Mixing Conditions 68 262 Inequalities 71 263 Limit Theorems for aMixing Processes 74 264 A Central Limit Theorem for Nonparametric Regression 76 271 Proof of Theorem 25 i 78 272 Proof of Proposition 23 i 79 274 Proof of Theorem 210 80 275 Proof of Theorem 213 81 277 Proof of Theorem 222 84 28 Additional Bibliographical Notes 87 ARMA Modeling and Forecasting 89 32 The Best Linear PredictionPrewhitening 91 33 Maximum Likelihood Estimation 93 332 Asymptotic Properties 97 555 Confidence Intervals 99 341 Akaike Information Criterion 100 342 FPE Criterion for AR Modeling 102 343 Bayesian Information Criterion 103 344 Model Identification 104 35 Diagnostic Checking 110 353 Tests for Whiteness 111 36 A Real Data ExampleAnalyzing German Egg Prices 113 37 Linear Forecasting 117 372 Forecasting in AR Processes 118 373 Mean Squared Predictive Errors for AR Processes 119 374 Forecasting in ARMA Processes 120 Parametric Nonlinear Time Series Models 124 411 Threshold Autoregressive Models 126 412 Estimation and Model Identification 131 413 Tests for Linearity 134 414 Case Studies with Canadian Lynx Data 136 42 ARCH and GARCH Models 143 422 Basic Properties of GARCH Processes 147 423 Estimation 156 424 Asymptotic Properties of Conditional MLEs 161 425 Bootstrap Confidence Intervals 163 426 Testing for the ARCH Effect 165 427 ARCH Modeling of Financial Data 168 Modeling SP 500 Index Returns 171 429 Stochastic Volatility Models 179 43 Bilinear Models 181 431 A Simple Example 182 432 Markovian Representation 184 433 Probabilistic Properties 185 434 Maximum Likelihood Estimation 189 44 Additional Bibliographical notes 191 Nonparametric Density Estimation 193 52 Kernel Density Estimation 194 53 Windowing and Whitening 197 54 Bandwidth Selection 199 55 Boundary Correction 202 56 Asymptotic Results 204 57 ComplementsProof of Theorem 53 211 58 Bibliographical Notes 212 Smoothing in Time Series 215 622 Moving Averages 217 623 Kernel Smoothing 218 624 Variations of Kernel Smoothers 220 625 Filtering 221 626 Local Linear Smoothing 222 627 Other Smoothing Methods 224 629 Theoretical Aspects 225 63 Smoothing in the State Domain 228 632 Local Polynomial Fitting 230 633 Properties of the Local Polynomial Estimator 234 634 Standard Errors and Estimated Bias 241 635 Bandwidth Selection 243 64 Spline Methods 246 641 Polynomial Splines 247 642 Nonquadratic Penalized Splines 249 643 Smoothing Splines 251 65 Estimation of Conditional Densities 253 652 Asymptotic Properties 256
 721 Tapering 277 722 Smoothing the Periodogram 281 723 Prewhitening and Bias Reduction 282 73 Automatic Estimation of Spectral Density 283 731 LeastSquares Estimators and Bandwidth Selection 284 732 Local Maximum Likelihood Estimator 286 733 Confidence Intervals 289 74 Tests for White Noise 296 742 Generalized Likelihood Ratio Test 298 743 x²Test and the Adaptive Neyman Test 300 744 Other SmoothingBased Tests 302 745 Numerical Examples 303 75 Complements 304 752 Lemmas 305 753 Proof of Theorem 71 306 754 Proof of Theorem 72 307 76 Bibliographical Notes 310 Nonparametric Models 313 82 Multivariate Local Polynomial Regression 314 822 Multivariate Local Linear Regression 316 823 Multivariate Local Quadratic Regression 317 83 FunctionalCoefficient Autoregressive Model 318 833 Ergodicity 319 834 Estimation of Coefficient Functions 321 835 Selection of Bandwidth and ModelDependent Variable 322 836 Prediction 324 838 Sampling Properties 332 84 Adaptive FunctionalCoefficient Autoregressive Models 333 841 The Models 334 842 Existence and Identifiability 335 843 Profile LeastSquares Estimation 337 844 Bandwidth Selection 340 846 Implementation 341 847 Examples 343 848 Extensions 349 852 The Backfitting Algorithm 350 853 Projections and Average Surface Estimators 352 854 Estimability of Coefficient Functions 354 855 Bandwidth Selection 355 856 Examples 356 86 Other Nonparametric Models 364 861 TwoTerm Interaction Models 365 862 Partially Linear Models 366 863 SingleIndex Models 367 864 MultipleIndex Models 368 865 An Analysis of Environmental Data 371 87 Modeling Conditional Variance 374 871 Methods of Estimating Conditional Variance 375 872 Univariate Setting 376 873 FunctionalCoefficient Models 382 875 Product Models 384 882 Technical Conditions for Theorems 82 and 83 386 883 Preliminaries to the Proof of Theorem 83 387 884 Proof of Theorem 83 390 885 Proof of Theorem 84 392 886 Conditions of Theorem 85 394 887 Proof of Theorem 85 395 89 Bibliographical Notes 399 Model Validation 405 92 Generalized Likelihood Ratio Tests 406 922 Generalized Likelihood Ratio Test 408 923 Null Distributions and the Bootstrap 409 924 Power of the GLR Test 414 926 Nonparametric versus Nonparametric Models 415 927 Choice of Bandwidth 416 928 A Numerical Example 417 93 Tests on Spectral Densities 419 931 Relation with Nonparametric Regression 421 933 Other Nonparametric Methods 425 934 Tests Based on Rescaled Periodogram 427 94 Autoregressive versus Nonparametric Models 430 942 Additive Alternatives 434 95 Threshold Models versus VaryingCoefficient Models 437 96 Bibliographical Notes 439 Nonlinear Prediction 440 1012 Noise Amplification 444 1013 Sensitivity to Initial Values 445 1014 MultipleStep Prediction versus a OneStep Plugin Method 447 1015 Nonlinear versus Linear Prediction 448 102 Point Prediction 450 1022 An Example 451 103 Estimating Predictive Distributions 454 1031 Local Logistic Estimator 455 1032 Adjusted NadarayaWatson Estimator 456 1033 Bootstrap Bandwidth Selection 457 1034 Numerical Examples 458 1035 Asymptotic Properties 463 1036 Sensitivity to Initial Values A Conditional Distribution Approach 466 104 Interval Predictors and Predictive Sets 470 1041 MinimumLength Predictive Sets 471 1042 Estimation of MinimumLength Predictors 474 1043 Numerical Examples 476 105 Complements 482 106 Additional Bibliographical Notes 485 References 487 Author index 537 Subject index 545 Copyright

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