Introduction to Time Series Modeling  book cover
1st Edition

Introduction to Time Series Modeling

ISBN 9781584889212
Published April 21, 2010 by Chapman and Hall/CRC
314 Pages - 80 B/W Illustrations

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Book Description

In time series modeling, the behavior of a certain phenomenon is expressed in relation to the past values of itself and other covariates. Since many important phenomena in statistical analysis are actually time series and the identification of conditional distribution of the phenomenon is an essential part of the statistical modeling, it is very important and useful to learn fundamental methods of time series modeling. Illustrating how to build models for time series using basic methods, Introduction to Time Series Modeling covers numerous time series models and the various tools for handling them.

The book employs the state-space model as a generic tool for time series modeling and presents convenient recursive filtering and smoothing methods, including the Kalman filter, the non-Gaussian filter, and the sequential Monte Carlo filter, for the state-space models. Taking a unified approach to model evaluation based on the entropy maximization principle advocated by Dr. Akaike, the author derives various methods of parameter estimation, such as the least squares method, the maximum likelihood method, recursive estimation for state-space models, and model selection by the Akaike information criterion (AIC). Along with simulation methods, he also covers standard stationary time series models, such as AR and ARMA models, as well as nonstationary time series models, including the locally stationary AR model, the trend model, the seasonal adjustment model, and the time-varying coefficient AR model.

With a focus on the description, modeling, prediction, and signal extraction of times series, this book provides basic tools for analyzing time series that arise in real-world problems. It encourages readers to build models for their own real-life problems.

Table of Contents

Introduction and Preparatory Analysis
Time Series Data
Classification of Time Series
Objectives of Time Series Analysis
Preprocessing of Time Series
Organization of This Book

The Covariance Function
The Distribution of Time Series and Stationarity
The Autocovariance Function of Stationary Time Series
Estimation of the Autocovariance Function
Multivariate Time Series and Scatterplots
Cross-Covariance Function and Cross-Correlation Function

The Power Spectrum and the Periodogram
The Power Spectrum
The Periodogram
Averaging and Smoothing of the Periodogram
Computational Method of Periodogram
Computation of the Periodogram by Fast Fourier Transform

Statistical Modeling
Probability Distributions and Statistical Models
K-L Information and the Entropy Maximization Principle
Estimation of the K-L Information and Log-Likelihood
Estimation of Parameters by the Maximum Likelihood Method
Akaike Information Criterion (AIC)
Transformation of Data

The Least Squares Method
Regression Models and the Least Squares Method
Householder Transformation Method
Selection of Order by AIC
Addition of Data and Successive Householder Reduction
Variable Selection by AIC

Analysis of Time Series Using ARMA Models
ARMA Model
The Impulse Response Function
The Autocovariance Function
The Relation between AR Coefficients and the PARCOR
The Power Spectrum of the ARMA Process
The Characteristic Equation
The Multivariate AR Model

Estimation of an AR Model
Fitting an AR Model
Yule–Walker Method and Levinson’s Algorithm
Estimation of an AR Model by the Least Squares Method
Estimation of an AR Model by the PARCOR Method
Large Sample Distribution of the Estimates
Yule–Walker Method for MAR Model
Least Squares Method for MAR Model

The Locally Stationary AR Model
Locally Stationary AR Model
Automatic Partitioning of the Time Interval
Precise Estimation of a Change Point

Analysis of Time Series with a State-Space Model
The State-Space Model
State Estimation via the Kalman Filter
Smoothing Algorithms
Increasing Horizon Prediction of the State
Prediction of Time Series
Likelihood Computation and Parameter Estimation for a Time Series Model
Interpolation of Missing Observations

Estimation of the ARMA Model
State-Space Representation of the ARMA Model
Initial State of an ARMA Model
Maximum Likelihood Estimate of an ARMA Model
Initial Estimates of Parameters

Estimation of Trends
The Polynomial Trend Model
Trend Component Model—Model for Probabilistic Structural Changes
Trend Model

The Seasonal Adjustment Model
Seasonal Component Model
Standard Seasonal Adjustment Model
Decomposition Including an AR Component
Decomposition Including a Trading-Day Effect

Time-Varying Coefficient AR Model
Time-Varying Variance Model
Time-Varying Coefficient AR Model
Estimation of the Time-Varying Spectrum
The Assumption on System Noise for the Time-Varying Coefficient AR Model
Abrupt Changes of Coefficients

Non-Gaussian State-Space Model
Necessity of Non-Gaussian Models
Non-Gaussian State-Space Models and State Estimation
Numerical Computation of the State Estimation Formula
Non-Gaussian Trend Model
A Time-Varying Variance Model
Applications of Non-Gaussian State-Space Model

The Sequential Monte Carlo Filter
The Nonlinear Non-Gaussian State-Space Model and Approximations of Distributions
Monte Carlo Filter
Monte Carlo Smoothing Method
Nonlinear Smoothing

Generation of Uniform Random Numbers
Generation of Gaussian White Noise
Simulation Using a State-Space Model
Simulation with Non-Gaussian Model

Appendix A: Algorithms for Nonlinear Optimization
Appendix B: Derivation of Levinson’s Algorithm
Appendix C: Derivation of the Kalman Filter and Smoother Algorithms
Appendix D: Algorithm for the Monte Carlo Filter


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Genshiro Kitagawa is the Director-General of the Institute of Statistical Mathematics in Tokyo, Japan.


My first reaction before opening this book was if there is a market for yet another book on the subject. However, my skepticism disappeared fast once I started reading the book. … The content of the book is well chosen … I would strongly recommend this book for readers who want to have a first glance to the notion of time series analysis and modelling. It is also a valuable book for teaching a first course on time series modelling both for graduate and/or undergraduate students.
Journal of Time Series Analysis, Volume 32, May 2011

This book provides an introduction to time series analysis with emphasis on the state space approach. It reflects the extensive experience and significant contributions of the author to non-linear and non-Gaussian modeling. … The material from Chapter 8 on is worth reading by anybody wishing to extend their range of time series tools. … This is a valuable book, especially with its broad and accessible introduction of models in the state space framework.
Statistics in Medicine, 2011, 30

… What distinguishes this book from comparable introductory texts is the use of state space modeling. Along with this come a number of valuable tools for recursive filtering and smoothing including the Kalman filter, as well as non-Gaussian and sequential Monte Carlo filters. … a useful reference for the application of state space modeling to time series.
MAA Reviews, October 2010