Time Series Analysis and Its Applications: With R Examples (Third

202 Pages·2011·4.8 MB·English

Springer Texts in Statistics

Series Editors
G. Casella
S. Fienberg
I. Olkin

For other titles published in this series, go to
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Robert H. Shumway • David S. Stoffer
Time Series Analysis and
Its Applications
With R Examples
Third edition
Prof. Robert H. Shumway Prof. David S. Stoffer
Department of Statistics Department of Statistics
University of California University of Pittsburgh
Davis, California Pittsburgh, Pennsylvania
USA USA

ISSN 1431 -875X
ISBN 978-1-4419-7864-6 e-ISBN 978-1-4419-7865-3
DOI 10.1007/978-1-4419-7865-3
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To my wife, Ruth, for her support and joie de vivre, and to the
memory of my thesis adviser, Solomon Kullback.
R.H.S.
To my family and friends, who constantly remind me what is
important.
D.S.S.

Preface to the Third Edition
The goals of this book are to develop an appreciation for the richness and
versatilityofmoderntimeseriesanalysisasatoolforanalyzingdata,andstill
maintainacommitmenttotheoreticalintegrity,asexemplifiedbytheseminal
worksofBrillinger(1975)andHannan(1970)andthetextsbyBrockwelland
Davis(1991)andFuller(1995).Theadventofinexpensivepowerfulcomputing
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 (1970). 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
graduatelevelsoverthepastdecade.Ourexperienceisthatanundergraduate
course can be accessible to students with a background in regression analysis
andmayinclude§1.1–§1.6,§2.1–§2.3,theresultsandnumericalpartsof§3.1–
§3.9,andbrieflytheresultsandnumericalpartsof§4.1–§4.6.Attheadvanced
undergraduate or master’s level, where the students have some mathematical
statistics background, more detailed coverage of the same sections, with the
inclusionof§2.4andextratopicsfromChapter5orChapter6canbeusedas
a one-semester course. Often, the extra topics are chosen by the students ac-
cording to their interests. Finally, a two-semester upper-level graduate course
for mathematics, statistics, and engineering graduate students can be crafted
byaddingselectedtheoreticalappendices.Fortheupper-levelgraduatecourse,
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 (1991), the
classic entry at this level.
Themajordifferencebetweenthisthirdeditionofthetextandthesecond
edition is that we provide R code for almost all of the numerical examples. In
addition,weprovideanRsupplementforthetextthatcontainsthedataand
scriptsinacompressedfilecalledtsa3.rda;thesupplementisavailableonthe
website for the third edition, http://www.stat.pitt.edu/stoffer/tsa3/,
viii Preface to the Third Edition
or one of its mirrors. On the website, we also provide the code used in each
example so that the reader may simply copy-and-paste code directly into R.
Specific details are given in Appendix R and on the website for the text.
Appendix R is new to this edition, and it includes a small R tutorial as well
asprovidingareferenceforthedatasetsandscriptsincludedintsa3.rda.So
there is no misunderstanding, we emphasize the fact that this text is about
time series analysis, not about R. R code is provided simply to enhance the
exposition by making the numerical examples reproducible.
Wehavetried,wherepossible,tokeeptheproblemsetsinordersothatan
instructormayhaveaneasytimemovingfromthesecondeditiontothethird
edition. However, some of the old problems have been revised and there are
somenewproblems.Also,someofthedatasetshavebeenupdated.Weadded
one section in Chapter 5 on unit roots and enhanced some of the presenta-
tions throughout the text. The exposition on state-space modeling, ARMAX
models,and(multivariate)regressionwithautocorrelatederrorsinChapter6
have been expanded. In this edition, we use standard R functions as much as
possible, but we use our own scripts (included in tsa3.rda) 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.
WethankJohnKimmel,ExecutiveEditor,SpringerStatistics,forhisguid-
ance in the preparation and production of this edition of the text. We are
grateful to Don Percival, University of Washington, for numerous suggestions
thatledtosubstantialimprovementtothepresentationinthesecondedition,
andconsequentlyinthisedition.WethankDougWiens,UniversityofAlberta,
for help with some of the R code in Chapters 4 and 7, and for his many sug-
gestions 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 the 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 Robert H. Shumway
Pittsburgh, PA David S. Stoffer
September 2010
Contents
Preface to the Third Edition................................... vii
1 Characteristics of Time Series ............................. 1
1.1 Introduction ............................................ 1
1.2 The Nature of Time Series Data........................... 3
1.3 Time Series Statistical Models ............................ 11
1.4 Measures of Dependence: Autocorrelation and
Cross-Correlation........................................ 17
1.5 Stationary Time Series ................................... 22
1.6 Estimation of Correlation................................. 28
1.7 Vector-Valued and Multidimensional Series ................. 33
Problems ................................................... 39
2 Time Series Regression and Exploratory Data Analysis .... 47
2.1 Introduction ............................................ 47
2.2 Classical Regression in the Time Series Context ............. 48
2.3 Exploratory Data Analysis................................ 57
2.4 Smoothing in the Time Series Context ..................... 70
Problems ................................................... 78
3 ARIMA Models ........................................... 83
3.1 Introduction ........................................... 83
3.2 Autoregressive Moving Average Models .................... 84
3.3 Difference Equations..................................... 97
3.4 Autocorrelation and Partial Autocorrelation ................102
3.5 Forecasting ............................................108
3.6 Estimation .............................................121
3.7 Integrated Models for Nonstationary Data .................141
3.8 Building ARIMA Models ................................144
3.9 Multiplicative Seasonal ARIMA Models ....................154
Problems ...................................................162