Nonlinear Time Series: Theory, Methods and Applications with R Examples

Preface

Frequently Used Notation

Part I  Foundations
    
Chapter 1  Linear Models
          Stochastic processes
          The covariance world
               Second-order stationary processes
               Spectral representation
               Wold decomposition
          Linear processes
               What are linear Gaussian processes?
               ARMA models
               Prediction
               Estimation
          The multivariate cases
               Time domain
               Frequency domain
          Numerical examples
          Exercises

Chapter 2  Linear Gaussian State Space Models
          Model basics
          Filtering, smoothing, and forecasting
          Maximum likelihood estimation
               Newton–Raphson
               EM algorithm
          Smoothing splines and the Kalman smoother
          Asymptotic distribution of the MLE
          Missing data modifications
          Structural component models
          State-space models with correlated errors
               ARMAX models
               Regression with autocorrelated errors
          Exercises

Chapter 3  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
               Integer valued models
               Generalized linear models
          Numerical examples
          Exercises

Chapter 4  Stochastic Recurrence Equations
          The Scalar Case
               Strict stationarity
               Weak stationarity
               GARCH(1, 1)
          The Vector Case
               Strict stationarity
               Weak stationarity
               GARCH(p, q)
          Iterated random function
               Strict stationarity
               Weak stationarity
          Exercises


Part II  Markovian Models

Chapter 5  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
               Metropolis-Hastings algorithm
               Gibbs sampling
          Exercises

Chapter 6  Stability and Convergence
          Uniform ergodicity
               Total variation distance
               Dobrushin coefficient
               The Doeblin condition
               Examples
          V-geometric ergodicity
               V-total variation distance
               V-Dobrushin coefficient
               Drift and minorization conditions
               Examples
          Some proofs
          Endnotes
          Exercises

Chapter 7  Sample Paths and Limit Theorems
          Law of large numbers
               Dynamical system and ergodicity
               Markov chain ergodicity
          Central limit theorem
          Deviation inequalities for additive functionals
               Rosenthal type inequality
               Concentration inequality
          Some proofs
          Exercises

Chapter 8  Inference for Markovian Models
          Likelihood inference
          Consistency and asymptotic normality of the MLE
               Consistency
               Asymptotic normality
          Observation-driven models
          Bayesian inference
          Some proofs
          Endnotes
          Exercises


Part III  State Space and Hidden Markov Models

Chapter 9  Non-Gaussian and Nonlinear State Space Models
          Definitions and basic properties
               Discrete-valued state space HMM
               Continuous-valued state-space models
               Conditionally Gaussian linear state-space models
               Switching processes with Markov regimes
          Filtering and smoothing
               Discrete-valued state-space HMM
               Continuous-valued state-space HMM
          Endnotes
          Exercises

Chapter 10  Particle Filtering
          Importance sampling
          Sequential importance sampling
          Sampling importance resampling
               Algorithm description
               Resampling techniques
          Particle filter
               Sequential importance sampling
               Auxiliary sampling
          Convergence of the particle filter
               Exponential deviation inequalities
               Time-uniform bounds
          Endnotes
          Exercises

Chapter 11  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
               Exponential deviation inequality
               Asymptotic normality
               Time uniform bounds
          Endnotes
          Exercises

Chapter 12  Inference for Nonlinear State Space Models
          Monte Carlo maximum likelihood estimation
               Particle approximation of the likelihood function
               Particle stochastic gradient
               Particle Monte Carlo EM algorithms
               Particle stochastic approximation EM
          Bayesian analysis
               Gaussian linear state space models
               Gibbs sampling for NLSS model
               Particle marginal Markov chain Monte Carlo
               Particle Gibbs algorithm
          Endnotes
          Exercises

Chapter 13  Asymptotics of the MLE for NLSS
          Strong consistency of the MLE
               Forgetting the initial distribution
               Approximation by conditional likelihood
               Strong consistency
               Identifiability of mixture densities
          Asymptotic normality
               Convergence of the observed information
               Limit distribution of the MLE
          Endnotes
          Exercises


Part IV  Appendices

Appendix Some Mathematical Background
          Some measure theory
          Some probability theory

Appendix Martingales
          Definitions and elementary properties
          Limits theorems

Appendix Stochastic Approximation
          Robbins–Monro algorithm: Elementary results
          Stochastic gradient
          Stepsize selection and averaging
          The Kiefer–Wolfowitz procedure

Appendix Data Augmentation
          The EM algorithm in the incomplete data model
          The Fisher and Louis identities
          Monte Carlo EM algorithm
               Stochastic approximation EM
          Convergence of the EM algorithm
          Convergence of the MCEM algorithm
               Convergence of perturbed dynamical systems
               Convergence of the MCEM algorithm

References

Index