Nonlinear Time Series: Nonparametric and Parametric Methods

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Springer Science & Business Media, Nov 23, 2007 - Mathematics - 553 pages
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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

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
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Page 520 - Approximation of a-stable probability densities using finite Gaussian mixtures," Proc. EUSIPCO'98 , Sep. 1998. 12. JP Nolan, "Multivariate stable distributions: approximation, estimation, simulation and identification," in A Practical Guide to Heavy Tails, RJ Adler, RE Feldman, and MS Taqqu, eds., Boston: Birkhauser, 1998.
Page 517 - CH (2000). Interactive tree-structured regression via principal Hessian directions. Journal of the American Statistical Association, 95, 547-560.
Page x - This work was partially supported by the Engineering and Physical Sciences Research Council of the UK, Grant Nr.

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