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Peter J. Brockwell Richard A. Davis Department of Statistics Department of Statistics Colorado State University Colorado State University Fort Collins, CO 80523 Fort Collins, CO 80523 USA USA pjbrock@stat.colostate.edu rdavis@stat.colostate.edu Editorial Board George Casella Stephen Fienberg Ingram Olkin Department of Statistics Department of Statistics Department of Statistics Griffin-Floyd Hall Carnegie Mellon University Stanford University University of Florida Pittsburgh, PA 15213-3890 Stanford, CA 94305 P.O. Box 118545 Gainesville, FL 32611-8545 USA USA USA Library of Congress Cataloging-in-Publication Data Brockwell, Peter J. Introduction to time series and forecasting / Peter J. Brockwell and Richard A. Davis.—2nd ed. p. cm. — (Springer texts in statistics) Includes bibliographical references and index. ISBN 0-387-95351-5 (alk. paper) 1. Time-series analysis. I. Davis, Richard A. II. Title. III. Series. QA280.B757 2002 519.5 ′ 5—dc21 2001049262 Printed on acid-free paper. © 2002, 1996 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publishers (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by MaryAnn Brickner; manufacturing supervised by Joe Quatela. Typeset by The Bartlett Press, Inc., Marietta, GA. Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 ISBN 0-387-95351-5 SPIN 10850334 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Disclaimer: This eBook does not include the ancillary media that was packaged with the original printed version of the book.
Page 1 and 2: Introduction to Time Series and For
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Page 10: To Pam and Patti
Page 14: Preface This book is aimed at the r
Page 18: Contents Preface vii 1. Introductio
Page 22: Contents xi 6.3. Unit Roots in Time
Page 26: Contents xiii B. Statistical Comple
Page 30: 1 Introduction 1.1 Examples of Time
Page 34: Figure 1-2 Results of the all-star
Page 38: Figure 1-5 Population of the U.S.A.
Page 42: 1.3 Some Simple Time Series Models
Page 50: Figure 1-8 Population of the U.S.A.
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1.4 Stationary Models and the Autoc
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1.5 Estimation and Elimination of T
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Figure 1-20 Smoothing with a low-pa
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Figure 1-22 Strike data smoothed by
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Figure 1-25 The estimated seasonal
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1.6 Testing the Estimated Noise Seq
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1.6 Testing the Estimated Noise Seq
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Figure 1-28 The sample autocorrelat
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Problems 41 d. Under the conditions
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Problems 43 1.13. Find a filter of
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2 Stationary 2.1 Basic Properties P
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2.1 Basic Properties 47 with corres
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2.1 Basic Properties 49 and σ 2 θ
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2.2 Linear Processes 2.2 Linear Pro
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2.2 Linear Processes 53 where ψj
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2.3 Introduction to ARMA Processes
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2.4 Properties of the Sample Mean a
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Figure 2-2 The sample autocorrelati
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2.5 Forecasting Stationary Time Ser
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2.6 The Wold Decomposition 77 Apply
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Problems 79 functions are also auto
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Problems 81 2.15. Suppose that {Xt,
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3 ARMA 3.1 ARMA(p, q) Processes Mod
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3.1 ARMA(p, q) Processes 85 Existen
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3.1 ARMA(p, q) Processes 87 {πj} g
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3.2 The ACF and PACF of an ARMA(p,
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Figure 3-3 The model ACF of the AR(
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Figure 3-5 Time series of the overs
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3.3 Forecasting ARMA Processes 101
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Problems 109 3.6. Show that the two
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4 Spectral Analysis 4.1 Spectral De
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4.1 Spectral Densities 113 1 2πN
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4.1 Spectral Densities 115 κ is an
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Figure 4-1 A sample path of size 10
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4.1 Spectral Densities 119 where {Z
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4.2 The Periodogram 121 ACF -0.5 0.
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4.2 The Periodogram 123 Now e1,...,
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4.2 The Periodogram 125 estimates i
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4.3 Time-Invariant Linear Filters 1
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4.3 Time-Invariant Linear Filters 1
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Figure 4-12 The transfer function D
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4.4 The Spectral Density of an ARMA
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Problems 135 4.5. If {Xt} and {Yt}
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5 Modeling and Forecasting with ARM
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5.1 Preliminary Estimation 139 of t
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5.1 Preliminary Estimation 141 Larg
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5.1 Preliminary Estimation 143 larg
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5.1 Preliminary Estimation 145 PACF
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5.1 Preliminary Estimation 147 Whil
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5.1 Preliminary Estimation 149 Exam
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5.1 Preliminary Estimation 151 Defi
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5.1 Preliminary Estimation 153 Rema
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5.1 Preliminary Estimation 155 Havi
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5.1 Preliminary Estimation 157 with
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5.2 Maximum Likelihood Estimation 1
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Figure 5-5 The rescaled residuals a
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5.4 Forecasting 5.4 Forecasting 167
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5.5 Order Selection 5.5 Order Selec
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5.5 Order Selection 171 Table 5.2
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5.5 Order Selection 173 It can be s
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Problems 175 for the mean-corrected
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Problems 177 that the mean is known
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6 Nonstationary and Seasonal Time S
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6.1 ARIMA Models for Nonstationary
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Figure 6-4 199 observations of the
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Figure 6-7 200 observations of the
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6.2 Identification Techniques 187 6
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Figure 6-11 The Australian red wine
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6.2 Identification Techniques 191 P
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6.3 Unit Roots in Time Series Model
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6.4 Forecasting ARIMA Models 199 of
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6.4 Forecasting ARIMA Models 201 wh
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6.5 Seasonal ARIMA Models 203 6.5 S
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Figure 6-15 The ACF of the model 6.
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6.5 Seasonal ARIMA Models 207 ACF -
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6.5 Seasonal ARIMA Models 209 and s
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6.6 Regression with ARMA Errors 211
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Problems Figure 6-19 The difference
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Problems 221 6.9. Repeat Problem 6.
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7 Multivariate Time Series 7.1 Exam
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7.1 Examples 225 and a natural esti
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Figure 7-3 The sample ACF ˆρ22 of
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Figure 7-6 The sample correlations
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7.2 Second-Order Properties of Mult
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7.2 Second-Order Properties of Mult
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7.3 Estimation of the Mean and Cova
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7.3 Estimation of the Mean and Cova
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Figure 7-7 The sample correlations
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7.4 Multivariate ARMA Processes 241
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7.4 Multivariate ARMA Processes 243
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7.5 Best Linear Predictors of Secon
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7.6 Modeling and Forecasting with M
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7.7 Cointegration 255 Example 7.7.1
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Problems 257 and derive the error c
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8 State-Space Models 8.1 State-Spac
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8.1 State-Space Representations 261
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8.2 The Basic Structural Model 263
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Figure 8-2 Sample ACF of the series
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8.3 State-Space Representation of A
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8.3 State-Space Representation of A
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8.4 The Kalman Recursions 271 the v
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8.4 The Kalman Recursions 273 Kalma
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8.4 The Kalman Recursions 275 where
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8.5 Estimation For State-Space Mode
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Figure 8-5 The one-step predictors
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8.6 State-Space Models with Missing
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8.6 State-Space Models with Missing
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8.7 The EM Algorithm 8.7 The EM Alg
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8.7 The EM Algorithm 291 Example 8.
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8.8 Generalized State-Space Models
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Figure 8-8 Goals scored by England
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Problems Problems 311 (see Problem
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Problems 313 can be expressed as Y
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Problems 315 and p(x2|y1) g(x2; y1
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9 Forecasting Techniques 9.1 The AR
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9.1 The ARAR Algorithm 319 and choo
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9.1 The ARAR Algorithm 321 9.1.4 Ap
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9.2 The Holt-Winters Algorithm 323
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Figure 9-2 The data set DEATHS.TSM
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Figure 9-3 The data set DEATHS.TSM
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Figure 9-4 The first 132 values of
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10Further Topics 10.1 Transfer Func
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10.1 Transfer Function Models 333 1
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10.1 Transfer Function Models 335 (
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Figure 10-2 The sample correlation
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10.1 Transfer Function Models 339 I
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10.2 Intervention Analysis 341 form
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10.3 Nonlinear Models 10.3 Nonlinea
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Figure 10-5 A sequence generated by
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10.3 Nonlinear Models 347 lation at
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10.3 Nonlinear Models 349 is a part
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Figure 10-7 A realization of the p
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10.3 Nonlinear Models 353 process {
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10.3 Nonlinear Models 355 Compariso
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10.4 Continuous-Time Models 357 whe
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10.4 Continuous-Time Models 359 i
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10.5 Long-Memory Models 361 and Cov
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10.5 Long-Memory Models 363 The spe
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Problems Figure 10-13 The minimum a
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Problems 367 c. For p ≥ 1, show t
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A Random Variables and Probability
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A.1 Distribution Functions and Expe
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A.1 Distribution Functions and Expe
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A.2 Random Vectors 375 The probabil
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A.3 The Multivariate Normal Distrib
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A.3 The Multivariate Normal Distrib
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Problems Problems 381 A.1. Let X ha
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B Statistical B.1 Least Squares Est
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B.1 Least Squares Estimation 385 Th
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B.2 Maximum Likelihood Estimation 3
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B.4 Hypothesis Testing 389 Example
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B.4 Hypothesis Testing 391 B.3.1, w
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C Mean C.1 The Cauchy Criterion Squ
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D An ITSM Tutorial D.1 Getting Star
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D.2 Preparing Your Data for Modelin
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D.2 Preparing Your Data for Modelin
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Figure D-2 The logged AIRPASS.TSM s
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D.3 Finding a Model for Your Data 4
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Figure D-4 The sample ACF of the tr
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D.4 Testing Your Model D.4 Testing
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D.4 Testing Your Model 413 frequenc
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D.5 Prediction D.5 Prediction 415 t
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D.6 Model Properties 417 of differe
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Figure D-12 The PACF of the model i
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D.7 Multivariate Time Series 421 us
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References Akaike, H. (1969), Fitti
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References 425 Dempster, A.P., Lair
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References 427 Mood, A.M., Graybill
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A accidental deaths (DEATHS.TSM), 3
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estimation of missing values in an
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polynomial fitting, 28 population o
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