# Nonlinear Time Series

## Theory, Methods and Applications with R Examples

© 2014 – Chapman and Hall/CRC

551 pages | 50 B/W Illus.

Hardback: 9781466502253
pub: 2014-01-06
\$102.95
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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.

### Reviews

"This book is very suitable for mathematicians requiring a very rigorous and complete introduction to nonlinear time series and their applications in several fields."

Zentralblatt MATH 1306

"This book focuses on theory and methods, with applications in mind. It is quite theory-heavy, with many rigorously established theoretical results. …It is also very timely and covers many recent developments in nonlinear time series analysis… readers can get a very up-to-date view of the current developments in nonlinear time series analysis from this book."

—Journal of the American Statistical Association, December 2014

"… the book will definitely help readers who are very mathematically inclined and keen on rigour and interested in further pursuing the probabilistic aspects of nonlinear time series. I have no doubt the book will be useful and timely, and I have no hesitation in recommending the book … ."

—T. Subba Rao, Journal of Time Series Analysis, 2014

FOUNDATIONS

Linear Models

Stochastic Processes

The Covariance World

Linear Processes

The Multivariate Cases

Numerical Examples

Exercises

Linear 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 Models

Nonlinear 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 MODELS

Markov 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 Convergence

Uniform Ergodicity

V-Geometric Ergodicity

Some Proofs

Endnotes

Exercises

Sample Paths and Limit Theorems

Law of Large Numbers

Central Limit Theorem

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 Models

Definitions 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