Table of Content
Preface
1 INTRODUCTION AND SOME BASIC CONCEPTS
1.1 Linearity and Gaussianity
1.2 Examples of Nonlinear Time Series
1.3 Initial Data Analysis
1.3.1 Skewness, kurtosis, and normality
1.3.2 Kendall’s (partial) tau
1.3.3 Mutual information coefficient
1.3.4 Recurrence plot
1.3.5 Directed scatter plot
1.4 Summary, Terms and Concepts
1.5 Additional Bibliographical Notes
1.6 Data and Software References
Exercises
2 CLASSIC NONLINEAR MODELS
2.1 The General Univariate Nonlinear Model
2.1.1 Volterra series expansions
2.1.2 State-dependent model formulation
2.2 Bilinear Models
2.3 Exponential ARMA Model
2.4 Random Coefficient AR Model
2.5 Nonlinear MA Model
2.6 Threshold Models
2.6.1 General threshold ARMA (TARMA) model
2.6.2 Self-exciting threshold ARMA model
2.6.3 Continuous SETAR model
2.6.4 Multivariate thresholds
2.6.5 Asymmetric ARMA model
2.6.6 Nested SETARMA model
2.7 Smooth Transition Models
2.8 Nonlinear non-Gaussian Models
2.8.1 Newer exponential autoregressive models
2.8.2 Product autoregressive model
2.9 Artificial Neural Network Models
2.9.1 AR neural network model
2.9.2 ARMA neural network model
2.9.3 Local global neural network model
2.9.4 Neuro-coefficient STAR model
2.10 Markov Switching Models
2.11 Application: An AR-NN model for EEG Recordings
2.12 Summary, Terms and Concepts
2.13 Additional Bibliographical Notes
2.14 Data and Software References
Appendix
2.A Impulse Response Functions
2.B Acronyms in Threshold Modeling
Exercises
3 PROBABILISTIC PROPERTIES
3.1 Strict Stationarity
3.2 Second-order Stationarity
3.3 Application: Nonlinear AR-GARCH model
3.4 Dependence and Geometric Ergodicity
3.4.1 Mixing coefficients
3.4.2 Geometric ergodicity
3.5 Invertibility
3.5.1 Global
3.5.2 Local
3.6 Summary, Terms and Concepts
3.7 Additional Bibliographical Notes
3.8 Software References
Appendix
3.A Vector and Matrix Norms
3.B Spectral Radius of a Matrix
Exercises
4 FREQUENCY-DOMAIN TESTS
4.1 Bispectrum
4.2 The Subba Rao–Gabr Tests
4.2.1 Testing for Gaussianity
4.2.2 Testing for linearity
4.2.3 Discussion
4.3 Hinich’s Tests
4.3.1 Testing for linearity
4.3.2 Testing for Gaussianity
4.3.3 Discussion
4.4 Related Tests
4.4.1 Goodness-of-fit tests
4.4.2 Maximal test statistics for linearity
4.4.3 Bootstrapped-based tests
4.4.4 Discussion
4.5 A MSFE-Based Linearity Test
4.6 Which Test to Use?
4.7 Application: A Comparison of Linearity Tests
4.8 Summary, Terms and Concepts
4.9 Additional Bibliographical Notes
4.10 Software References
Exercises
5 TIME-DOMAIN LINEARITY TESTS
5.1 Lagrange Multiplier Tests
5.2 Likelihood Ratio Tests
5.3 Wald Test
5.4 Tests Based on a Second-order Volterra Expansion
5.5 Tests Based on Arranged Autoregressions
5.6 Nonlinearity vs. Specific Nonlinear Alternatives
5.7 Summary, Terms and Concepts
5.8 Additional Bibliographical Notes
5.9 Software References
Appendix
5.A Percentiles LR-SETAR Test Statistic
5.B Summary of Size and Power Studies
Exercises
6 MODEL ESTIMATION, SELECTION, AND CHECKING
6.1 Model Estimation
6.1.1 Quasi maximum likelihood estimator
6.1.2 Conditional least squares estimator
6.1.3 Iteratively weighted least squares
6.2 Model Selection Tools
6.2.1 Kullback–Leibler information
6.2.2 The AIC, AICc, and AICu rules
6.2.3 Generalized information criterion: The GIC rule
6.2.4 Bayesian approach: The BIC rule
6.2.5 Minimum descriptive length principle
6.2.6 Model selection in threshold models
6.3 Diagnostic Checking
6.3.1 Pearson residuals
6.3.2 Quantile residuals
6.4 Application: TARSO Model of a Water Table
6.5 Summary, Terms and Concepts
6.6 Additional Bibliographical Notes
6.7 Data and Software References
Exercises
7 TESTS FOR SERIAL INDEPENDENCE
7.1 Null Hypothesis
7.2 Distance Measures and Dependence Functionals
7.2.1 Correlation integral
7.2.2 Quadratic distance
7.2.3 Density-based measures
7.2.4 Distribution-based measures
7.2.5 Copula-based measures
7.3 Kernel-Based Tests
7.3.1 Density estimators
7.3.2 Copula estimators
7.3.3 Single-lag test statistics
7.3.4 Multiple-lag test statistics
7.3.5 Generalized spectral tests
7.3.6 Computing p-values
7.4 High-Dimensional Tests
7.4.1 BDS test statistic
7.4.2 Rank-based BDS test statistics
7.4.3 Distribution-based test statistics
7.4.4 Copula-based test statistics
7.4.5 A test statistic based on quadratic forms
7.5 Application: Canadian Lynx Data
7.6 Summary, Terms and Concepts
7.7 Additional Bibliographical Notes
7.8 Data and Software References
Appendix
7.A Kernel-based Density and Regression Estimation
7.B Copula Theory
7.C U- and V-statistics
Exercises
8 TIME-REVERSIBILITY
8.1 Preliminaries
8.2 Time-Domain Tests
8.2.1 A bicovariance-based test
8.2.2 A test based on the characteristic function
8.3 Frequency-Domain Tests
8.3.1 A bispectrum-based test
8.3.2 A trispectrum-based test
8.4 Other Nonparametric Tests
8.4.1 A copula-based tests for Markov chains
8.4.2 A kernel-based test
8.4.3 A sign test
8.5 Application: A Comparison of TR Tests
8.6 Summary, Terms and Concepts
8.7 Additional Bibliographical Notes
8.8 Software References
Exercises
9 SEMI- AND NONPARAMETRIC FORECASTING
9.1 Kernel-based Nonparametric Methods
9.1.1 Conditional mean, median, and mode
9.1.2 Single- and multi-stage quantile prediction
9.1.3 Conditional densities
9.1.4 Locally weighted regression
9.1.5 Conditional mean and variance
9.1.6 Model assessment and lag selection
9.2 Semiparametric Methods
9.2.1 ACE and AVAS
9.2.2 Projection pursuit regression
9.2.3 Multivariate adaptive regression splines (MARS)
9.2.4 Boosting
9.2.5 Functional-coefficient AR models
9.2.6 Single-index coefficient model
9.3 Summary, Terms and Concepts
9.4 Additional Bibliographical Notes
9.5 Data and Software References
Exercises
10 FORECASTING
10.1 Exact Least Squares Forecasting Methods
10.1.1 Nonlinear AR model
10.1.2 Self-exciting threshold ARMA model
10.2 Approximate Forecasting Methods
10.2.1 Monte Carlo
10.2.2 Bootstrap
10.2.3 Deterministic, naive, or skeleton
10.2.4 Empirical least squares
10.2.5 Normal forecasting error
10.2.6 Linearization
10.2.7 Dynamic estimation
10.3 Forecast Intervals and Regions
10.3.1 Preliminaries
10.3.2 Conditional percentiles
10.3.3 Conditional densities
10.4 Forecast Evaluation
10.4.1 Point forecast
10.4.2 Interval evaluation
10.4.3 Density evaluation
10.5 Forecast Combination
10.6 Summary, Terms and Concepts
10.7 Additional Bibliographical Notes
Exercises
11 VECTOR PARAMETRIC MODELS AND METHODS
11.1 General Multivariate Nonlinear Model
11.2 Vector Models
11.2.1 Bilinear models
11.2.2 General threshold ARMA (TARMA) model
11.2.3 VSETAR with multivariate thresholds
11.2.4 Threshold vector error correction
11.2.5 Vector smooth transition AR
11.2.6 Vector smooth transition error correction
11.2.7 Other vector nonlinear models
11.3 Time-Domain Linearity Tests
11.4 Testing Linearity vs. Specific Nonlinear Alternatives
11.5 Model Selection Tools
11.6 Diagnostic Checking
11.6.1 Quantile residuals
11.7 Forecasting
11.7.1 Point forecasts
11.7.2 Forecast evaluation
11.8 Application: Analysis of Icelandic River Flow Data
11.9 Summary, Terms and Concepts
11.10 Additional Bibliographical Notes
11.11 Data and Software References
Appendix
11.A Percentiles of the LR-VTAR Test Statistic
11.B Computing GIRFs
Exercises
12 VECTOR SEMI- AND NONPARAMETRIC METHODS
12.1 Nonparametric Methods
12.1.1 Conditional quantiles
12.1.2 Kernel-based forecasting
12.1.3 K-nearest neighbors
12.2 Semiparametric methods
12.2.1 PolyMARS
12.2.2 Projection pursuit regression
12.2.3 Vector functional-coefficient AR model
12.3 Frequency-Domain Tests
12.4 Lag Selection
12.5 Nonparametric Causality Testing
12.5.1 Preamble
12.5.2 A bivariate nonlinear causality test statistic
12.5.3 A modified bivariate causality test statistic
12.5.4 A multivariate causality test statistic
12.6 Summary, Terms and Concepts
12.7 Additional Bibliographical Notes
12.8 Data and Software References
Appendix
12.A Computing Multivariate Conditional Quantiles
12.B Percentiles of the R(l) Test Statistic
Exercises
References
Books about Nonlinear Time Series Analysis
Notation and Abbreviations
List of Pseudocode Algorithms
List of Examples
Subject index
1 INTRODUCTION AND SOME BASIC CONCEPTS
1.1 Linearity and Gaussianity
1.2 Examples of Nonlinear Time Series
1.3 Initial Data Analysis
1.3.1 Skewness, kurtosis, and normality
1.3.2 Kendall’s (partial) tau
1.3.3 Mutual information coefficient
1.3.4 Recurrence plot
1.3.5 Directed scatter plot
1.4 Summary, Terms and Concepts
1.5 Additional Bibliographical Notes
1.6 Data and Software References
Exercises
2 CLASSIC NONLINEAR MODELS
2.1 The General Univariate Nonlinear Model
2.1.1 Volterra series expansions
2.1.2 State-dependent model formulation
2.2 Bilinear Models
2.3 Exponential ARMA Model
2.4 Random Coefficient AR Model
2.5 Nonlinear MA Model
2.6 Threshold Models
2.6.1 General threshold ARMA (TARMA) model
2.6.2 Self-exciting threshold ARMA model
2.6.3 Continuous SETAR model
2.6.4 Multivariate thresholds
2.6.5 Asymmetric ARMA model
2.6.6 Nested SETARMA model
2.7 Smooth Transition Models
2.8 Nonlinear non-Gaussian Models
2.8.1 Newer exponential autoregressive models
2.8.2 Product autoregressive model
2.9 Artificial Neural Network Models
2.9.1 AR neural network model
2.9.2 ARMA neural network model
2.9.3 Local global neural network model
2.9.4 Neuro-coefficient STAR model
2.10 Markov Switching Models
2.11 Application: An AR-NN model for EEG Recordings
2.12 Summary, Terms and Concepts
2.13 Additional Bibliographical Notes
2.14 Data and Software References
Appendix
2.A Impulse Response Functions
2.B Acronyms in Threshold Modeling
Exercises
3 PROBABILISTIC PROPERTIES
3.1 Strict Stationarity
3.2 Second-order Stationarity
3.3 Application: Nonlinear AR-GARCH model
3.4 Dependence and Geometric Ergodicity
3.4.1 Mixing coefficients
3.4.2 Geometric ergodicity
3.5 Invertibility
3.5.1 Global
3.5.2 Local
3.6 Summary, Terms and Concepts
3.7 Additional Bibliographical Notes
3.8 Software References
Appendix
3.A Vector and Matrix Norms
3.B Spectral Radius of a Matrix
Exercises
4 FREQUENCY-DOMAIN TESTS
4.1 Bispectrum
4.2 The Subba Rao–Gabr Tests
4.2.1 Testing for Gaussianity
4.2.2 Testing for linearity
4.2.3 Discussion
4.3 Hinich’s Tests
4.3.1 Testing for linearity
4.3.2 Testing for Gaussianity
4.3.3 Discussion
4.4 Related Tests
4.4.1 Goodness-of-fit tests
4.4.2 Maximal test statistics for linearity
4.4.3 Bootstrapped-based tests
4.4.4 Discussion
4.5 A MSFE-Based Linearity Test
4.6 Which Test to Use?
4.7 Application: A Comparison of Linearity Tests
4.8 Summary, Terms and Concepts
4.9 Additional Bibliographical Notes
4.10 Software References
Exercises
5 TIME-DOMAIN LINEARITY TESTS
5.1 Lagrange Multiplier Tests
5.2 Likelihood Ratio Tests
5.3 Wald Test
5.4 Tests Based on a Second-order Volterra Expansion
5.5 Tests Based on Arranged Autoregressions
5.6 Nonlinearity vs. Specific Nonlinear Alternatives
5.7 Summary, Terms and Concepts
5.8 Additional Bibliographical Notes
5.9 Software References
Appendix
5.A Percentiles LR-SETAR Test Statistic
5.B Summary of Size and Power Studies
Exercises
6 MODEL ESTIMATION, SELECTION, AND CHECKING
6.1 Model Estimation
6.1.1 Quasi maximum likelihood estimator
6.1.2 Conditional least squares estimator
6.1.3 Iteratively weighted least squares
6.2 Model Selection Tools
6.2.1 Kullback–Leibler information
6.2.2 The AIC, AICc, and AICu rules
6.2.3 Generalized information criterion: The GIC rule
6.2.4 Bayesian approach: The BIC rule
6.2.5 Minimum descriptive length principle
6.2.6 Model selection in threshold models
6.3 Diagnostic Checking
6.3.1 Pearson residuals
6.3.2 Quantile residuals
6.4 Application: TARSO Model of a Water Table
6.5 Summary, Terms and Concepts
6.6 Additional Bibliographical Notes
6.7 Data and Software References
Exercises
7 TESTS FOR SERIAL INDEPENDENCE
7.1 Null Hypothesis
7.2 Distance Measures and Dependence Functionals
7.2.1 Correlation integral
7.2.2 Quadratic distance
7.2.3 Density-based measures
7.2.4 Distribution-based measures
7.2.5 Copula-based measures
7.3 Kernel-Based Tests
7.3.1 Density estimators
7.3.2 Copula estimators
7.3.3 Single-lag test statistics
7.3.4 Multiple-lag test statistics
7.3.5 Generalized spectral tests
7.3.6 Computing p-values
7.4 High-Dimensional Tests
7.4.1 BDS test statistic
7.4.2 Rank-based BDS test statistics
7.4.3 Distribution-based test statistics
7.4.4 Copula-based test statistics
7.4.5 A test statistic based on quadratic forms
7.5 Application: Canadian Lynx Data
7.6 Summary, Terms and Concepts
7.7 Additional Bibliographical Notes
7.8 Data and Software References
Appendix
7.A Kernel-based Density and Regression Estimation
7.B Copula Theory
7.C U- and V-statistics
Exercises
8 TIME-REVERSIBILITY
8.1 Preliminaries
8.2 Time-Domain Tests
8.2.1 A bicovariance-based test
8.2.2 A test based on the characteristic function
8.3 Frequency-Domain Tests
8.3.1 A bispectrum-based test
8.3.2 A trispectrum-based test
8.4 Other Nonparametric Tests
8.4.1 A copula-based tests for Markov chains
8.4.2 A kernel-based test
8.4.3 A sign test
8.5 Application: A Comparison of TR Tests
8.6 Summary, Terms and Concepts
8.7 Additional Bibliographical Notes
8.8 Software References
Exercises
9 SEMI- AND NONPARAMETRIC FORECASTING
9.1 Kernel-based Nonparametric Methods
9.1.1 Conditional mean, median, and mode
9.1.2 Single- and multi-stage quantile prediction
9.1.3 Conditional densities
9.1.4 Locally weighted regression
9.1.5 Conditional mean and variance
9.1.6 Model assessment and lag selection
9.2 Semiparametric Methods
9.2.1 ACE and AVAS
9.2.2 Projection pursuit regression
9.2.3 Multivariate adaptive regression splines (MARS)
9.2.4 Boosting
9.2.5 Functional-coefficient AR models
9.2.6 Single-index coefficient model
9.3 Summary, Terms and Concepts
9.4 Additional Bibliographical Notes
9.5 Data and Software References
Exercises
10 FORECASTING
10.1 Exact Least Squares Forecasting Methods
10.1.1 Nonlinear AR model
10.1.2 Self-exciting threshold ARMA model
10.2 Approximate Forecasting Methods
10.2.1 Monte Carlo
10.2.2 Bootstrap
10.2.3 Deterministic, naive, or skeleton
10.2.4 Empirical least squares
10.2.5 Normal forecasting error
10.2.6 Linearization
10.2.7 Dynamic estimation
10.3 Forecast Intervals and Regions
10.3.1 Preliminaries
10.3.2 Conditional percentiles
10.3.3 Conditional densities
10.4 Forecast Evaluation
10.4.1 Point forecast
10.4.2 Interval evaluation
10.4.3 Density evaluation
10.5 Forecast Combination
10.6 Summary, Terms and Concepts
10.7 Additional Bibliographical Notes
Exercises
11 VECTOR PARAMETRIC MODELS AND METHODS
11.1 General Multivariate Nonlinear Model
11.2 Vector Models
11.2.1 Bilinear models
11.2.2 General threshold ARMA (TARMA) model
11.2.3 VSETAR with multivariate thresholds
11.2.4 Threshold vector error correction
11.2.5 Vector smooth transition AR
11.2.6 Vector smooth transition error correction
11.2.7 Other vector nonlinear models
11.3 Time-Domain Linearity Tests
11.4 Testing Linearity vs. Specific Nonlinear Alternatives
11.5 Model Selection Tools
11.6 Diagnostic Checking
11.6.1 Quantile residuals
11.7 Forecasting
11.7.1 Point forecasts
11.7.2 Forecast evaluation
11.8 Application: Analysis of Icelandic River Flow Data
11.9 Summary, Terms and Concepts
11.10 Additional Bibliographical Notes
11.11 Data and Software References
Appendix
11.A Percentiles of the LR-VTAR Test Statistic
11.B Computing GIRFs
Exercises
12 VECTOR SEMI- AND NONPARAMETRIC METHODS
12.1 Nonparametric Methods
12.1.1 Conditional quantiles
12.1.2 Kernel-based forecasting
12.1.3 K-nearest neighbors
12.2 Semiparametric methods
12.2.1 PolyMARS
12.2.2 Projection pursuit regression
12.2.3 Vector functional-coefficient AR model
12.3 Frequency-Domain Tests
12.4 Lag Selection
12.5 Nonparametric Causality Testing
12.5.1 Preamble
12.5.2 A bivariate nonlinear causality test statistic
12.5.3 A modified bivariate causality test statistic
12.5.4 A multivariate causality test statistic
12.6 Summary, Terms and Concepts
12.7 Additional Bibliographical Notes
12.8 Data and Software References
Appendix
12.A Computing Multivariate Conditional Quantiles
12.B Percentiles of the R(l) Test Statistic
Exercises
References
Books about Nonlinear Time Series Analysis
Notation and Abbreviations
List of Pseudocode Algorithms
List of Examples
Subject index