Nonlinear Time Series Analysis

RUEY S. TSAY, PHD, is H.G.B. Alexander Professor of Econometrics and Statistics at The University of Chicago Booth School of Business. He is a fellow of the American Statistical Association and the Institute of Mathematical Statistics.Dr. Tsay is author of Analysis of Financial Time Series, Multivariate Time Series Analysis, and An Introduction to Analysis of Financial Data with R all published by Wiley.RONG CHEN, PHD, is Distinguished Professor of Statistics and Director of the Master programs in Financial Statistics and Risk Management and in Data Science at Rutgers University. He is a fellow of the American Statistical Association and the Institute of Mathematical Statistics.

• Preface xiii1 Why Should We Care About Nonlinearity? 11.1 Some Basic Concepts 21.2 Linear Time Series 31.3 Examples of Nonlinear Time Series 31.4 Nonlinearity Tests 201.4.1 Nonparametric Tests 211.4.2 Parametric Tests 311.5 Exercises 38References 392 Univariate Parametric Nonlinear Models 412.1 A General Formulation 412.1.1 Probability Structure 422.2 Threshold Autoregressive Models 432.2.1 A Two-regime TAR Model 442.2.2 Properties of Two-regime TAR(1) Models 452.2.3 Multiple-regime TAR Models 482.2.4 Estimation of TAR Models 502.2.5 TAR Modeling 522.2.6 Examples 552.2.7 Predictions of TAR Models 622.3 Markov Switching Models 632.3.1 Properties of Markov Switching Models 662.3.2 Statistical Inference of the State Variable 662.3.3 Estimation of Markov Switching Models 692.3.4 Selecting the Number of States 752.3.5 Prediction of Markov Switching Models 752.3.6 Examples 762.4 Smooth Transition Autoregressive Models 922.5 Time-varying Coefficient Models 992.5.1 Functional Coefficient AR Models 992.5.2 Time-varying Coefficient AR Models 1042.6 Appendix: Markov Chains 1112.7 Exercises 114References 1163 Univariate Nonparametric Models 1193.1 Kernel Smoothing 1193.2 Local Conditional Mean 1253.3 Local Polynomial Fitting 1293.4 Splines 1343.4.1 Cubic and B-Splines 1383.4.2 Smoothing Splines 1413.5 Wavelet Smoothing 1453.5.1 Wavelets 1453.5.2 The Wavelet Transform 1473.5.3 Thresholding and Smoothing 1503.6 Nonlinear Additive Models 1583.7 Index Model and Sliced Inverse Regression 1643.8 Exercises 169References 1704 Neural Networks, Deep Learning, and Tree-based Methods 1734.1 Neural Networks 1734.1.1 Estimation or Training of Neural Networks 1764.1.2 An Example 1794.2 Deep Learning 1814.2.1 Deep Belief Nets 1824.2.2 Demonstration 1844.3 Tree-based Methods 1954.3.1 Decision Trees 1954.3.2 Random Forests 2124.4 Exercises 214References 2155 Analysis of Non-Gaussian Time Series 2175.1 Generalized Linear Time Series Models 2185.1.1 Count Data and GLARMA Models 2205.2 Autoregressive Conditional Mean Models 2295.3 Martingalized GARMA Models 2325.4 Volatility Models 2345.5 Functional Time Series 2455.5.1 Convolution FAR models 2485.5.2 Estimation of CFAR Models 2515.5.3 Fitted Values and Approximate Residuals 2535.5.4 Prediction 2535.5.5 Asymptotic Properties 2545.5.6 Application 254Appendix: Discrete Distributions for Count Data 2605.6 Exercises 261References 2636 State Space Models 2656.1 A General Model and Statistical Inference 2666.2 Selected Examples 2696.2.1 Linear Time Series Models 2696.2.2 Time Series with Observational Noises 2716.2.3 Time-varying Coefficient Models 2726.2.4 Target Tracking 2736.2.5 Signal Processing in Communications 2796.2.6 Dynamic Factor Models 2836.2.7 Functional and Distributional Time Series 2846.2.8 Markov Regime Switching Models 2896.2.9 Stochastic Volatility Models 2906.2.10 Non-Gaussian Time Series 2916.2.11 Mixed Frequency Models 2916.2.12 Other Applications 2926.3 Linear Gaussian State Space Models 2936.3.1 Filtering and the Kalman Filter 2936.3.2 Evaluating the likelihood function 2956.3.3 Smoothing 2976.3.4 Prediction and Missing Data 2996.3.5 Sequential Processing 3006.3.6 Examples and R Demonstrations 3006.4 Exercises 325References 3277 Nonlinear State Space Models 3357.1 Linear and Gaussian Approximations 3357.1.1 Kalman Filter for Linear Non-Gaussian Systems 3367.1.2 Extended Kalman Filters for Nonlinear Systems 3367.1.3 Gaussian Sum Filters 3387.1.4 The Unscented Kalman Filter 3397.1.5 Ensemble Kalman Filters 3417.1.6 Examples and R implementations 3427.2 Hidden Markov Models 3517.2.1 Filtering 3517.2.2 Smoothing 3527.2.3 The Most Likely State Path: the Viterbi Algorithm 3557.2.4 Parameter Estimation: the Baum–Welch Algorithm 3567.2.5 HMM Examples and R Implementation 3587.3 Exercises 371References 3728 Sequential Monte Carlo 3758.1 A Brief Overview of Monte Carlo Methods 3768.1.1 General Methods of Generating Random Samples 3788.1.2 Variance Reduction Methods 3848.1.3 Importance Sampling 3878.1.4 Markov Chain Monte Carlo 3988.2 The SMC Framework 4028.3 Design Issue I: Propagation 4108.3.1 Proposal Distributions 4118.3.2 Delay Strategy (Lookahead) 4158.4 Design Issue II: Resampling 4218.4.1 The Priority Score 4228.4.2 Choice of Sampling Methods in Resampling 4238.4.3 Resampling Schedule 4258.4.4 Benefits of Resampling 4268.5 Design Issue III: Inference 4288.6 Design Issue IV: Marginalization and the Mixture Kalman Filter 4298.6.1 Conditional Dynamic Linear Models 4298.6.2 Mixture Kalman Filters 4308.7 Smoothing with SMC 4338.7.1 Simple Weighting Approach 4338.7.2 Weight Marginalization Approach 4348.7.3 Two-filter Sampling 4368.8 Parameter Estimation with SMC 4388.8.1 Maximum Likelihood Estimation 4388.8.2 Bayesian Parameter Estimation 4418.8.3 Varying Parameter Approach 4418.9 Implementation Considerations 4428.10 Examples and R Implementation 4448.10.1 R Implementation of SMC: Generic SMC and Resampling Methods 4448.10.2 Tracking in a Clutter Environment 4498.10.3 Bearing-only Tracking with Passive Sonar 4668.10.4 Stochastic Volatility Models 4718.10.5 Fading Channels as Conditional Dynamic Linear Models 4788.11 Exercises 486References 487Index 493