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Time_Series_Analysis_and_its_Applications_with_R_Examples_4th_ed.pdf
Contents
1 Characteristics of Time Series
1.1 The Nature of Time Series Data
1.2 Time Series Statistical Models
1.3 Measures of Dependence
1.4 Stationary Time Series
1.5 Estimation of Correlation
1.6 Vector-Valued and Multidimensional Series
Problems
2 Time Series Regression & Exploratory Data Analysis
2.1 Classical Regression in the Time Series Context
2.2 Exploratory Data Analysis
2.3 Smoothing in the Time Series Context
Problems
3 ARIMA Models
3.1 Autoregressive Moving Average Models
3.2 Difference Equations
3.3 Autocorrelation and Partial Autocorrelation
3.4 Forecasting
3.5 Estimation
3.6 Integrated Models for Nonstationary Data
3.7 Building ARIMA Models
3.8 Regression with Autocorrelated Errors
3.9 Multiplicative Seasonal ARIMA Models
Problems
4 Spectral Analysis & Filtering
4.1 Cyclical Behavior and Periodicity
4.2 The Spectral Density
4.3 Periodogram and Discrete Fourier Transform
4.4 Nonparametric Spectral Estimation
4.5 Parametric Spectral Estimation
4.6 Multiple Series and Cross-Spectra
4.7 Linear Filters
4.8 Lagged Regression Models
4.9 Signal Extraction and Optimum Filtering
4.10 Spectral Analysis of Multidimensional Series
Problems
5 Additional Time Domain Topics
5.1 Long Memory ARMA and Fractional Differencing
5.2 Unit Root Testing
5.3 GARCH Models
5.4 Threshold Models
5.5 Lagged Regression and Transfer Function Modeling
5.6 Multivariate ARMAX Models
Problems
6 State Space Models
6.1 Linear Gaussian Model
6.2 Filtering, Smoothing, and Forecasting
6.3 Maximum Likelihood Estimation
6.4 Missing Data Modifications
6.5 Structural Models: Signal Extraction and Forecasting
6.6 State-Space Models with Correlated Errors
6.6.1 ARMAX Models
6.6.2 Multivariate Regression with Autocorrelated Errors
6.7 Bootstrapping State Space Models
6.8 Smoothing Splines and the Kalman Smoother
6.9 Hidden Markov Models and Switching Autoregression
6.10 Dynamic Linear Models with Switching
6.11 Stochastic Volatility
6.12 Bayesian Analysis of State Space Models
Problems
7 Statistical Methods in Frequency Domain
7.1 Introduction
7.2 Spectral Matrices and Likelihood Functions
7.3 Regression for Jointly Stationary Series
7.4 Regression with Deterministic Inputs
7.5 Random Coefficient Regression
7.6 Analysis of Designed Experiments
7.7 Discriminant and Cluster Analysis
7.8 Principal Components and Factor Analysis
7.9 The Spectral Envelope
Problems
Large Sample Theory
Convergence Modes
Central Limit Theorems
Mean & Autocorrelation Functions
Time Domain Theory
Hilbert Spaces & Projection Theorem
Causal Conditions for ARMA Models
Large Sample Distribution of AR Conditional Least Squares Estimators
The Wold Decomposition
Spectral Domain Theory
Spectral Representation Theorems
Large Sample Distribution of Smoothed Periodogram
Complex Multivariate Normal Distribution
Integration
Spectral Analysis as Principal Component Analysis
Parametric Spectral Estimation
R Supplement
First Things first
astsa
Start
Time Series Primer
Refs
Index
Robert H. Shumway • David S. Stoffer
Time Series Analysis
and Its Applications
With R Examples
Fourth Edition
123
Robert H. Shumway
Department of Statistics
University of California, Davis
Davis, CA, USA
David S. Stoffer
Department of Statistics
University of Pittsburgh
Pittsburgh, PA, USA
ISSN 1431-875X
Springer Texts in Statistics
ISBN 978-3-319-52451-1
DOI 10.1007/978-3-319-52452-8
ISSN 2197-4136 (electronic)
ISBN 978-3-319-52452-8 (eBook)
Library of Congress Control Number: 2017930675
© Springer International Publishing AG 1999, 2012, 2016, 2017
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface to the Fourth Edition
The fourth edition follows the general layout of the third edition but includes some
modernization of topics as well as the coverage of additional topics. The preface to
the third edition—which follows—still applies, so we concentrate on the differences
between the two editions here. As in the third edition, R code for each example
is given in the text, even if the code is excruciatingly long. Most of the examples
with seemingly endless coding are in the latter chapters. The R package for the
text, astsa, is still supported and details may be found in Appendix R. The global
temperature deviation series have been updated to 2015 and are included in the newest
version of the package; the corresponding examples and problems have been updated
accordingly.
Chapter 1 of this edition is similar to the previous edition, but we have included
the definition of trend stationarity and the concept of prewhitening when using cross-
correlation. The New York Stock Exchange data set, which focused on an old financial
crisis, was replaced with a more current series of the Dow Jones Industrial Average,
which focuses on a newer financial crisis. In Chap. 2, we rewrote some of the
regression review, changing the smoothing examples from the mortality data example
to the Southern Oscillation Index and finding El Niño. We also expanded on the lagged
regression example and carried it on to Chap. 3.
In Chap. 3, we removed normality from definition of ARMA models; while the
assumption is not necessary for the definition, it is essential for inference and pre-
diction. We added a section on regression with ARMA errors and the corresponding
problems; this section was previously in Chap. 5. Some of the examples have been
modified and we added some examples in the seasonal ARMA section. Finally, we
included a discussion of lagged regression with autocorrelated errors.
In Chap. 4, we improved and added some examples. The idea of modulated
series is discussed using the classic star magnitude data set. We moved some of the
filtering section forward for easier access to information when needed. We removed
the reliance on spec.pgram (from the stats package) to mvspec (from the astsa
package) so we can avoid having to spend pages explaining the quirks of spec.pgram,
which tended to take over the narrative. The section on wavelets was removed because
there are so many accessible texts available. The spectral representation theorems are
discussed in a little more detail using examples based on simple harmonic processes.
The general layout of Chap. 5 and of Chap. 7 is the same, although we have
revised some of the examples. As previously mentioned, we moved regression with
ARMA errors to Chap. 3.
Chapter 6 sees the biggest change in this edition. We have added a section on
smoothing splines, and a section on hidden Markov models and switching autore-
gressions. The Bayesian section is completely rewritten and is on linear Gaussian
state space models only. The nonlinear material in the previous edition is removed
because it was old, and the newer material is in Douc, Moulines, and Stoffer [53].
Many of the examples have been rewritten to make the chapter more accessible.
The appendices are similar, with some minor changes to Appendix A and
Appendix B. We added material to Appendix C, including a discussion of Riemann–
Stieltjes and stochastic integration, a proof of the fact that the spectra of autoregressive
processes are dense in the space of spectral densities, and a proof of the fact that spec-
tra are approximately the eigenvalues of the covariance matrix of a stationary process.
We tweaked, rewrote, improved, or revised some of the exercises, but the overall
ordering and coverage is roughly the same. And, of course, we moved regression with
ARMA errors problems to Chap. 3 and removed the Chap. 4 wavelet problems. The
exercises for Chap. 6 have been updated accordingly to reflect the new and improved
version of the chapter.
Davis, CA, USA
Pittsburgh, PA, USA
December 2016
Robert H. Shumway
David S. Stoffer
Preface to the Third Edition
The goals of this book are to develop an appreciation for the richness and versatility
of modern time series analysis as a tool for analyzing data, and still maintain a
commitment to theoretical integrity, as exemplified by the seminal works of Brillinger
[33] and Hannan [86] and the texts by Brockwell and Davis [36] and Fuller [66]. The
advent of inexpensive powerful computing has provided both real data and new
software that can take one considerably beyond the fitting of simple time domain
models, such as have been elegantly described in the landmark work of Box and
Jenkins [30]. This book is designed to be useful as a text for courses in time series on
several different levels and as a reference work for practitioners facing the analysis of
time-correlated data in the physical, biological, and social sciences.
We have used earlier versions of the text at both the undergraduate and gradu-
ate levels over the past decade. Our experience is that an undergraduate course can
be accessible to students with a background in regression analysis and may include
Sects. 1.1–1.5, Sects. 2.1–2.3, the results and numerical parts of Sects. 3.1–3.9,
and briefly the results and numerical parts of Sects. 4.1–4.4. At the advanced un-
dergraduate or master’s level, where the students have some mathematical statistics
background, more detailed coverage of the same sections, with the inclusion of extra
topics from Chaps. 5 or 6, can be used as a one-semester course. Often, the extra
topics are chosen by the students according to their interests. Finally, a two-semester
upper-level graduate course for mathematics, statistics, and engineering graduate stu-
dents can be crafted by adding selected theoretical appendices. For the upper-level
graduate course, we should mention that we are striving for a broader but less rigorous
level of coverage than that which is attained by Brockwell and Davis [36], the classic
entry at this level.
The major difference between this third edition of the text and the second edition is
that we provide R code for almost all of the numerical examples. An R package called
astsa is provided for use with the text; see Sect. R.2 for details. R code is provided
simply to enhance the exposition by making the numerical examples reproducible.
We have tried, where possible, to keep the problem sets in order so that an
instructor may have an easy time moving from the second edition to the third edition.
However, some of the old problems have been revised and there are some new
problems. Also, some of the data sets have been updated. We added one section in
Chap. 5 on unit roots and enhanced some of the presentations throughout the text. The
exposition on state-space modeling, ARMAX models, and (multivariate) regression
with autocorrelated errors in Chap. 6 have been expanded. In this edition, we use
standard R functions as much as possible, but we use our own scripts (included in
astsa) when we feel it is necessary to avoid problems with a particular R function;
these problems are discussed in detail on the website for the text under R Issues.
We thank John Kimmel, Executive Editor, Springer Statistics, for his guidance
in the preparation and production of this edition of the text. We are grateful to Don
Percival, University of Washington, for numerous suggestions that led to substantial
improvement to the presentation in the second edition, and consequently in this
edition. We thank Doug Wiens, University of Alberta, for help with some of the
R code in Chaps. 4 and 7, and for his many suggestions for improvement of the
exposition. We are grateful for the continued help and advice of Pierre Duchesne,
University of Montreal, and Alexander Aue, University of California, Davis. We also
thank the many students and other readers who took the time to mention typographical
errors and other corrections to the first and second editions. Finally, work on this
edition was supported by the National Science Foundation while one of us (D.S.S.)
was working at the Foundation under the Intergovernmental Personnel Act.
Davis, CA, USA
Pittsburgh, PA, USA
September 2010
Robert H. Shumway
David S. Stoffer
Contents
v
vii
1
2
8
15
19
26
33
38
Preface to the Fourth Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Characteristics of Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 The Nature of Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Time Series Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Measures of Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Stationary Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Estimation of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Vector-Valued and Multidimensional Series . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time Series Regression and Exploratory Data Analysis . . . . . . . . . . . . .
2.1 Classical Regression in the Time Series Context . . . . . . . . . . . . . . . . .
2.2 Exploratory Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Smoothing in the Time Series Context . . . . . . . . . . . . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
75
3.1 Autoregressive Moving Average Models . . . . . . . . . . . . . . . . . . . . . . .
88
3.2 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Autocorrelation and Partial Autocorrelation. . . . . . . . . . . . . . . . . . . . .
94
3.4 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6 Integrated Models for Nonstationary Data . . . . . . . . . . . . . . . . . . . . . . 131
3.7 Building ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.8 Regression with Autocorrelated Errors . . . . . . . . . . . . . . . . . . . . . . . . 142
3.9 Multiplicative Seasonal ARIMA Models . . . . . . . . . . . . . . . . . . . . . . . 145
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
45
45
54
65
70
2
3
4
5
6
7
Spectral Analysis and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.1 Cyclical Behavior and Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.2 The Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.3 Periodogram and Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . 179
4.4 Nonparametric Spectral Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.5 Parametric Spectral Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.6 Multiple Series and Cross-Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.7 Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.8 Lagged Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.9 Signal Extraction and Optimum Filtering . . . . . . . . . . . . . . . . . . . . . . . 222
4.10 Spectral Analysis of Multidimensional Series . . . . . . . . . . . . . . . . . . . 226
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Additional Time Domain Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.1 Long Memory ARMA and Fractional Differencing . . . . . . . . . . . . . . 241
5.2 Unit Root Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.3 GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.4 Threshold Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
5.5 Lagged Regression and Transfer Function Modeling . . . . . . . . . . . . . 266
5.6 Multivariate ARMAX Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
6.1 Linear Gaussian Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.2 Filtering, Smoothing, and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
6.4 Missing Data Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
6.5 Structural Models: Signal Extraction and Forecasting . . . . . . . . . . . . 318
6.6 State-Space Models with Correlated Errors . . . . . . . . . . . . . . . . . . . . . 321
6.6.1 ARMAX Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.6.2 Multivariate Regression with Autocorrelated Errors . . . . . . . 324
6.7 Bootstrapping State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
6.8 Smoothing Splines and the Kalman Smoother . . . . . . . . . . . . . . . . . . . 333
6.9 Hidden Markov Models and Switching Autoregression . . . . . . . . . . . 336
6.10 Dynamic Linear Models with Switching . . . . . . . . . . . . . . . . . . . . . . . 348
6.11 Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
6.12 Bayesian Analysis of State Space Models . . . . . . . . . . . . . . . . . . . . . . 367
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Statistical Methods in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . 385
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
7.2 Spectral Matrices and Likelihood Functions . . . . . . . . . . . . . . . . . . . . 388
7.3 Regression for Jointly Stationary Series . . . . . . . . . . . . . . . . . . . . . . . . 390
7.4 Regression with Deterministic Inputs
. . . . . . . . . . . . . . . . . . . . . . . . . 399
7.5 Random Coefficient Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
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