Time Series Analysis, Fourth Edition

765 Pages·2008·6.954 MB·English

by  George E. P. Box, Gwilym M. Jenkins, Gregory C. Reinsel(auth.)

Time Series Analysis
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Time Series Analysis
Forecasting and Control
FOURTH EDITION
GEORGE E. P. BOX
GWILYM M. JENKINS
GREGORY C. REINSEL
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Box,GeorgeE.P.
Timeseriesanalysis:forecastingandcontrol/George E.P. Box, Gwilym M.
Jenkins, Gregory C. Reinsel. —4thed.
p.cm.
Includesindex.
ISBN-978-0-470-27284-8(cloth)
1. Time-seriesanalysis. 2. Predicitiontheory. 3. Transferfunctions. 4.
Feedbackcontrolsystems—Mathematicalmodels. I. Jenkins, Gwilym M. II.
Reinsel,GregoryC.III.Title
QA280.B672008
519.5(cid:1)5—dc22
2007044569
PrintedintheUnitedStatesofAmerica
10987654321
To the memory of
Gwilym M. Jenkins Gregory C. Reinsel
Contents
Preface to the Fourth Edition xxi
Preface to the Third Edition xxiii
1 Introduction 1
1.1 Five Important Practical Problems, 2
1.1.1 Forecasting Time Series, 2
1.1.2 Estimation of Transfer Functions, 3
1.1.3 Analysis of Effects of Unusual Intervention
Events to a System, 4
1.1.4 Analysis of Multivariate Time Series, 5
1.1.5 Discrete Control Systems, 5
1.2 Stochastic and Deterministic Dynamic Mathematical
Models, 7
1.2.1 Stationary and Nonstationary Stochastic Models
for Forecasting and Control, 7
1.2.2 Transfer Function Models, 12
1.2.3 Models for Discrete Control Systems, 14
1.3 Basic Ideas in Model Building, 16
1.3.1 Parsimony, 16
1.3.2 Iterative Stages in the Selection of a Model, 17
Part One Stochastic Models and Their Forecasting 19
2 Autocorrelation Function and Spectrum of Stationary Processes 21
2.1 Autocorrelation Properties of Stationary Models, 21
2.1.1 Time Series and Stochastic Processes, 21
2.1.2 Stationary Stochastic Processes, 24
vii
viii CONTENTS
2.1.3 Positive Definiteness and the Autocovariance
Matrix, 25
2.1.4 Autocovariance and Autocorrelation
Functions, 29
2.1.5 Estimation of Autocovariance and
Autocorrelation Functions, 31
2.1.6 Standard Errors of Autocorrelation Estimates, 33
2.2 Spectral Properties of Stationary Models, 35
2.2.1 Periodogram of a Time Series, 35
2.2.2 Analysis of Variance, 37
2.2.3 Spectrum and Spectral Density Function, 38
2.2.4 Simple Examples of Autocorrelation and Spectral
Density Functions, 43
2.2.5 Advantages and Disadvantages of the
Autocorrelation and Spectral Density
Functions, 45
A2.1 Link between the Sample Spectrum and Autocovariance
Function Estimate, 45
3 Linear Stationary Models 47
3.1 General Linear Process, 47
3.1.1 Two Equivalent Forms for the Linear Process, 47
3.1.2 Autocovariance Generating Function of a Linear
Process, 50
3.1.3 Stationarity and Invertibility Conditions for a
Linear Process, 51
3.1.4 Autoregressive and Moving Average
Processes, 53
3.2 Autoregressive Processes, 55
3.2.1 Stationarity Conditions for Autoregressive
Processes, 55
3.2.2 Autocorrelation Function and Spectrum of
Autoregressive Processes, 57
3.2.3 First-Order Autoregressive (Markov) Process, 59
3.2.4 Second-Order Autoregressive Process, 61
3.2.5 Partial Autocorrelation Function, 66
3.2.6 Estimation of the Partial Autocorrelation
Function, 69
3.2.7 Standard Errors of Partial Autocorrelation
Estimates, 70
3.3 Moving Average Processes, 71
CONTENTS ix
3.3.1 Invertibility Conditions for Moving Average
Processes, 71
3.3.2 Autocorrelation Function and Spectrum of
Moving Average Processes, 72
3.3.3 First-Order Moving Average Process, 73
3.3.4 Second-Order Moving Average Process, 75
3.3.5 Duality Between Autoregressive and Moving
Average Processes, 78
3.4 Mixed Autoregressive–Moving Average Processes, 79
3.4.1 Stationarity and Invertibility Properties, 79
3.4.2 Autocorrelation Function and Spectrum of Mixed
Processes, 80
3.4.3 First-Order Autoregressive–First-Order Moving
Average Process, 82
3.4.4 Summary, 86
A3.1 Autocovariances, Autocovariance Generating Function, and
Stationarity Conditions for a General Linear Process, 86
A3.2 Recursive Method for Calculating Estimates of
Autoregressive Parameters, 89
4 Linear Nonstationary Models 93
4.1 Autoregressive Integrated Moving Average Processes, 93
4.1.1 Nonstationary First-Order Autoregressive
Process, 93
4.1.2 General Model for a Nonstationary Process
Exhibiting Homogeneity, 95
4.1.3 General Form of the Autoregressive Integrated
Moving Average Model, 100
4.2 Three Explicit Forms for The Autoregressive Integrated
Moving Average Model, 103
4.2.1 Difference Equation Form of the Model, 103
4.2.2 Random Shock Form of the Model, 104
4.2.3 Inverted Form of the Model, 111
4.3 Integrated Moving Average Processes, 114
4.3.1 Integrated Moving Average Process of
Order (0, 1, 1), 115
4.3.2 Integrated Moving Average Process of
Order (0, 2, 2), 119
4.3.3 General Integrated Moving Average Process of
Order (0,d,q), 123
A4.1 Linear Difference Equations, 125
A4.2 IMA(0, 1, 1) Process with Deterministic Drift, 131
x CONTENTS
A4.3 Arima Processes with Added Noise, 131
A4.3.1 Sum of Two Independent Moving Average
Processes, 132
A4.3.2 Effect of Added Noise on the General
Model, 133
A4.3.3 Example for an IMA(0, 1, 1) Process with Added
White Noise, 134
A4.3.4 Relation between the IMA(0, 1, 1) Process and a
Random Walk, 135
A4.3.5 Autocovariance Function of the General Model
with Added Correlated Noise, 135
5 Forecasting 137
5.1 Minimum Mean Square Error Forecasts and Their
Properties, 137
5.1.1 Derivation of the Minimum Mean Square Error
Forecasts, 139
5.1.2 Three Basic Forms for the Forecast, 141
5.2 Calculating and Updating Forecasts, 145
5.2.1 Convenient Format for the Forecasts, 145
5.2.2 Calculation of the ψ Weights, 147
5.2.3 Use of the ψ Weights in Updating the
Forecasts, 148
5.2.4 Calculation of the Probability Limits of the
Forecasts at Any Lead Time, 150
5.3 Forecast Function and Forecast Weights, 152
5.3.1 Eventual Forecast Function Determined by the
Autoregressive Operator, 152
5.3.2 Role of the Moving Average Operator in Fixing
the Initial Values, 153
5.3.3 Lead l Forecast Weights, 154
5.4 Examples of Forecast Functions and Their Updating, 157
5.4.1 Forecasting an IMA(0, 1, 1) Process, 157
5.4.2 Forecasting an IMA(0, 2, 2) Process, 160
5.4.3 Forecasting a General IMA(0, d, q) Process, 163
5.4.4 Forecasting Autoregressive Processes, 164
5.4.5 Forecasting a (1, 0, 1) Process, 167
5.4.6 Forecasting a (1, 1, 1) Process, 169
CONTENTS xi
5.5 Use of State-Space Model Formulation for Exact
Forecasting, 170
5.5.1 State-Space Model Representation for the
ARIMA Process, 170
5.5.2 Kalman Filtering Relations for Use
in Prediction, 171
5.5.3 Smoothing Relations in the State Variable
Model, 175
5.6 Summary, 177
A5.1 Correlations Between Forecast Errors, 180
A5.1.1 Autocorrelation Function of Forecast Errors at
Different Origins, 180
A5.1.2 Correlation Between Forecast Errors at the Same
Origin with Different Lead Times, 182
A5.2 Forecast Weights for Any Lead Time, 182
A5.3 Forecasting in Terms of the General Integrated Form, 185
A5.3.1 General Method of Obtaining the Integrated
Form, 185
A5.3.2 Updating the General Integrated Form, 187
A5.3.3 Comparison with the Discounted Least Squares
Method, 187
Part Two Stochastic Model Building 193
6 Model Identification 195
6.1 Objectives of Identification, 195
6.1.1 Stages in the Identification Procedure, 195
6.2 Identification Techniques, 196
6.2.1 Use of the Autocorrelation and Partial
Autocorrelation Functions in Identification, 196
6.2.2 Standard Errors for Estimated Autocorrelations
and Partial Autocorrelations, 198
6.2.3 Identification of Some Actual Time Series, 200
6.2.4 Some Additional Model Identification Tools, 208
6.3 Initial Estimates for the Parameters, 213
6.3.1 Uniqueness of Estimates Obtained from the
Autocovariance Function, 213
6.3.2 Initial Estimates for Moving Average
Processes, 213
6.3.3 Initial Estimates for Autoregressive
Processes, 215