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Introduction to Time Series Modeling

Genshiro Kitagawa
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Monographs on Statistics and Applied Probability 114
[Reviewed by
William J. Satzer
, on

A time series is a sequence of data points. Typically the data points are arranged according to the time at which they are collected, though certain other kinds of sequential data can be amenable to time series analysis. Usually the data are collected at uniform time intervals. The characteristic property of time series is that the data are not generated independently, and the nature of the dependence is of great interest. Often the data embody trends and have cyclic components. These too are of significant interest. Time series arise in essentially any discipline in which sequences of numerical data are collected. Applications range from medical (electrocardiograms and electroencephalograms) to business and economics (stock prices and interest rates) to meteorology (daily maximum temperatures and precipitation).

Time series analysis consists of methods for exploring sequential data in order to extract meaningful statistics and identify significant characteristics. The goals of the analysis generally include at least one of the following: development of models to understand the stochastic processes underlying the data, prediction of future events based on past events, application of control to influence future values of the data, and identification of persistent signals present in the data.

The current book focuses on the modeling of time series. In roughly the first half of the text, the author introduces the basic ideas and primary tools of time series analysis. A significant part of this is the treatment of autoregressive models, such as ARMA (autoregressive moving average) that represent a time series value as a linear combination of past values and white noise. The challenge with these is first to estimate the order of the AR model (how many past values to use), then to estimate the unknown coefficients. The author describes several approaches, including the standard one that uses the Yule-Walker method and Levinson’s algorithm.

Although both time-domain and frequency-domain approaches to time series modeling are commonly used, the author concentrates on the time domain. Just one chapter discusses spectral analysis and periodiograms. 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.

As an introduction to the subject, this book has an excessively theoretical emphasis. Although the author provides many graphs of interesting time series from a variety of sources, there are very few examples of computation using those data. Most students learn time series analysis best by applying the techniques they’re learning to real data sets, but they have very little opportunity to see that done here. By contrast, Time Series Analysis: Forecasting and Control by Box and Jenkins, a classic text in this field, uses a collection of standard time series data to great effect in order to illustrate individual analysis methods. Another important practical issue is that common errors in modeling and analyzing time series are not considered. Yet that is something that students need to see.

The current book is a useful reference for the application of state space modeling to time series. As an introductory textbook, however, it leaves much to be desired. 




Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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