Nonlinear Time Series Theory, Methods and Applications with R Examples

Designed for researchers and students, Nonlinear Times Series: Theory, Methods and Applications with R Examples familiarizes readers with the principles behind nonlinear time series models—without overwhelming them with difficult mathematical developments. By focusing on basic principles and theory, the authors give readers the background required to craft their own stochastic models, numerical methods, and software. They will also be able to assess the advantages and disadvantages of different approaches, and thus be able to choose the right methods for their purposes. The first part can be seen as a crash course on "classical" time series, with a special emphasis on linear state space models and detailed coverage of random coefficient autoregressions, both ARCH and GARCH models. The second part introduces Markov chains, discussing stability, the existence of a stationary distribution, ergodicity, limit theorems, and statistical inference. The book concludes with a self-contained account on nonlinear state space and sequential Monte Carlo methods. An elementary introduction to nonlinear state space modeling and sequential Monte Carlo, this section touches on current topics, from the theory of statistical inference to advanced computational methods. The book can be used as a support to an advanced course on these methods, or an introduction to this field before studying more specialized texts. Several chapters highlight recent developments such as explicit rate of convergence of Markov chains and sequential Monte Carlo techniques. And while the chapters are organized in a logical progression, the three parts can be studied independently. Statistics is not a spectator sport, so the book contains more than 200 exercises to challenge readers. These problems strengthen intellectual muscles strained by the introduction of new theory and go on to extend the theory in significant ways. The book helps readers hone their skills in nonlinear time series analysis and their applications.

FOUNDATIONSLinear ModelsStochastic Processes The Covariance World Linear Processes The Multivariate Cases Numerical Examples ExercisesLinear Gaussian State Space Models Model Basics Filtering, Smoothing, and Forecasting Maximum Likelihood Estimation Smoothing Splines and the Kalman Smoother Asymptotic Distribution of the MLE Missing Data Modifications Structural Component Models State-Space Models with Correlated Errors Exercises Beyond Linear ModelsNonlinear Non-Gaussian Data Volterra Series Expansion Cumulants and Higher-Order Spectra Bilinear Models Conditionally Heteroscedastic Models Threshold ARMA Models Functional Autoregressive Models Linear Processes with Infinite Variance Models for Counts Numerical Examples Exercises Stochastic Recurrence Equations The Scalar Case The Vector Case Iterated Random Function Exercises MARKOVIAN MODELSMarkov Models: Construction and Definitions Markov Chains: Past, Future and forgetfulness Kernels Homogeneous Markov Chain Canonical Representation Invariant Measures Observation-Driven Models Iterated Random Functions MCMC Methods Exercises Stability and ConvergenceUniform Ergodicity V-Geometric Ergodicity Some Proofs Endnotes Exercises Sample Paths and Limit Theorems Law of Large Numbers Central Limit Theorem Deviation Inequalities for Additive Functionals Some Proofs Exercises Inference for Markovian Models Likelihood Inference MLE: Consistency and Asymptotic Normality Observation-Driven Models Bayesian Inference Some Proofs Endnotes Exercises STATE SPACE AND HIDDEN MARKOV MODELS Non-Gaussian and Nonlinear State Space ModelsDefinitions and basic properties Filtering and smoothing Endnotes Exercises Particle Filtering Importance sampling Sequential importance sampling Sampling importance resampling Particle filter Convergence of the particle filter Endnotes Exercises Particle Smoothing Poor man’s Smoother Algorithm FFBSm Algorithm FFBSi Algorithm Smoothing Functionals Particle Independent Metropolis-Hastings Particle Gibbs Convergence of the FFBSm and FFBSi Algorithms Endnotes Exercises Inference for Nonlinear State Space Models Monte Carlo Maximum Likelihood Estimation Bayesian Analysis Endnotes Exercises Asymptotics of the MLE for NLSS Strong Consistency of the MLE Asymptotic Normality Endnotes Exercises APPENDICES Appendix A: Some Mathematical Background Appendix B: Martingales Appendix C: Stochastic Approximation Appendix D: Data Augmentation References