Time series analysis

619 Pages·2016·24.87 MB·English

TIME SERIES ANALYSIS
WILEY SERIES IN PROBABILITY AND STATISTICS
Established by WALTER A. SHEWHART and SAMUEL S. WILKS
Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice,
Geof H. Givens, Harvey Goldstein, Geert Molenberghs, David W. Scott,
Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg
Editors Emeriti: J. Stuart Hunter, Iain M. Johnstone, Joseph B. Kadane,
Jozef L. Teugels
A complete list of the titles in this series appears at the end of this volume.
TIME SERIES ANALYSIS
Wilfredo Palma
Pontificia Universidad Cat´olica de Chile
Copyright(cid:2)c 2016byJohnWiley&Sons,Inc.Allrightsreserved
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey
PublishedsimultaneouslyinCanada
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinany
formorbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise,
exceptaspermittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,without
eitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpaymentofthe
appropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers,
MA01923,[phone],fax[phone],oronthewebatwww.copyright.com.Requeststo
thePublisherforpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&
Sons,Inc.,111RiverStreet,Hoboken,NJ07030,[phone],fax[phone],oronlineat
http://www.wiley.com/go/permission.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheirbest
effortsinpreparingthisbook,theymakenorepresentationsorwarrantieswithrespecttothe
accuracyorcompletenessofthecontentsofthisbookandspecificallydisclaimanyimplied
warrantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreatedor
extendedbysalesrepresentativesorwrittensalesmaterials.Theadviceandstrategiescontained
hereinmaynotbesuitableforyoursituation.Youshouldconsultwithaprofessionalwhere
appropriate.Neitherthepublishernorauthorshallbeliableforanylossofprofitoranyother
commercialdamages,includingbutnotlimitedtospecial,incidental,consequential,orother
damages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,pleasecontact
ourCustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outsidetheUnited
Statesat(317)572-3993orfax(317)572-4002.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprint
maynotbeavailableinelectronicformats.FormoreinformationaboutWileyproducts,visitour
websiteatwww.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Palma,Wilfredo,1963-
Timeseriesanalysis/WilfredoPalma,PonticiaUniversidadCato´licadeChile.
pagescm
Includesbibliographicalreferencesandindexes.
ISBN978-1-118-63432-5(cloth)
1.Time-seriesanalysis.I.Title.
QA280.P3542016
519.5(cid:3)5–dc23
2015024282
PrintedintheUnitedStatesofAmerica
10987654321
CONTENTS
Preface xiii
Acknowledgments xvii
Acronyms xix
About the Companion Website xxv
1 Introduction 1
1.1 Time Series Data 2
1.1.1 Financial Data 2
1.1.2 Economic Data 6
1.1.3 Hydrological Data 6
1.1.4 Air Pollution 7
1.1.5 Transportation Data 9
1.1.6 Biomedical Time Series 9
1.1.7 Sociological Data 10
1.1.8 Energy Data 11
1.1.9 Climatic Data 12
1.2 Random Variables and Statistical Modeling 16
1.3 Discrete-Time Models 22
v
vi CONTENTS
1.4 Serial Dependence 22
1.5 Nonstationarity 25
1.6 Whiteness Testing 32
1.7 Parametric and Nonparametric Modeling 36
1.8 Forecasting 38
1.9 Time Series Modeling 38
1.10 Bibliographic Notes 39
Problems 40
2 Linear Processes 43
2.1 Definition 44
2.2 Stationarity 44
2.3 Invertibility 46
2.4 Causality 46
2.5 Representations of Linear Processes 46
2.5.1 Wold Decomposition 47
2.5.2 Autoregressive Representation 48
2.5.3 State Space Systems 48
2.6 Weak and Strong Dependence 49
2.7 ARMA Models 51
2.7.1 Invertibility of ARMA Processes 52
2.7.2 Simulated ARMA Processes 52
2.8 Autocovariance Function 56
2.9 ACF and Partial ACF Functions 57
2.9.1 Sample ACF 60
2.9.2 Partial ACF 63
2.10 ARFIMA Processes 64
2.10.1 Long-Memory Processes 64
2.10.2 Linear Representations 65
2.10.3 Autocovariance Function 66
2.10.4 Sample Mean 67
2.10.5 Partial Autocorrelations 67
2.10.6 Illustrations 68
2.11 Fractional Gaussian Noise 71
2.11.1 Sample Mean 72
2.12 Bibliographic Notes 72
Problems 72
CONTENTS vii
3 State Space Models 89
3.1 Introduction 90
3.2 Linear Dynamical Systems 92
3.2.1 Stability 92
3.2.2 Hankel Matrix 93
3.2.3 Observability 94
3.2.4 Controllability 94
3.2.5 Minimality 95
3.3 State space Modeling of Linear Processes 96
3.3.1 State Space Form to Wold Decomposition 96
3.3.2 Wold Decomposition to State Space Form 96
3.3.3 Hankel Matrix to State Space Form 96
3.4 State Estimation 97
3.4.1 State Predictor 98
3.4.2 State Filter 98
3.4.3 State Smoother 99
3.4.4 Missing Values 99
3.4.5 Additive Noise 105
3.4.6 Structural Models 110
3.4.7 Estimation of Future States 111
3.5 Exogenous Variables 113
3.6 Bibliographic Notes 114
Problems 114
4 Spectral Analysis 121
4.1 Time and Frequency Domains 122
4.2 Linear Filters 122
4.3 Spectral Density 123
4.4 Periodogram 125
4.5 Smoothed Periodogram 128
4.6 Examples 130
4.7 Wavelets 136
4.8 Spectral Representation 138
4.9 Time-Varying Spectrum 140
4.10 Bibliographic Notes 145
Problems 145
viii CONTENTS
5 Estimation Methods 151
5.1 Model Building 152
5.2 Parsimony 152
5.3 Akaike and Schwartz Information Criteria 153
5.4 Estimation of the Mean 153
5.5 Estimation of Autocovariances 154
5.6 Moment Estimation 155
5.7 Maximum-Likelihood Estimation 156
5.7.1 Cholesky Decomposition Method 156
5.7.2 Durbin-Levinson Algorithm 157
5.8 Whittle Estimation 157
5.9 State Space Estimation 160
5.10 Estimation of Long-Memory Processes 161
5.10.1 Autoregressive Approximations 162
5.10.2 Haslett-Raftery Method 162
5.10.3 A State Space Method 164
5.10.4 Moving-Average Approximations 165
5.10.5 Semiparametric Approach 168
5.10.6 Periodogram Regression 169
5.10.7 Rescaled Range Method 170
5.10.8 Variance Plots 171
5.10.9 Detrended Fluctuation Analysis 171
5.10.10A Wavelet-Based Method 174
5.10.11Computation of Autocovariances 177
5.11 Numerical Experiments 178
5.12 Bayesian Estimation 180
5.12.1 Markov Chain Monte Carlo Methods 181
5.12.2 Metropolis-Hastings Algorithm 181
5.12.3 Gibbs Sampler 182
5.13 Statistical Inference 184
5.14 Illustrations 189
5.15 Bibliographic Notes 193
Problems 194
6 Nonlinear Time Series 209
6.1 Introduction 210
6.2 Testing for Linearity 211
6.3 Heteroskedastic Data 212