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5.1 Preliminary Estimation 149 Example 5.1.4 The lake data estimated white noise variance and larger Gaussian likelihood. From the ratio of the estimated coefficient to (1.96× standard error) displayed by ITSM, we obtain the 95% confidence bounds for φ: 0.4371 ± 0.4371/2.1668 (0.2354, 0.6388). This series {Yt,t 1,...,98} has already been studied in Example 1.3.5. In this example we shall consider the problem of fitting an AR process directly to the data without first removing any trend component. A graph of the data was displayed in Figure 1.9. The sample ACF and PACF are shown in Figures 5.3 and 5.4, respectively. The sample PACF shown in Figure 5.4 strongly suggests fitting an AR(2) model to the mean-corrected data Xt Yt − 9.0041. After clicking on the blue preliminary estimation button of ITSM select Yes to subtract the sample mean from {Yt}. Then specify 2 for the AR order, 0 for the MA order, and Burg for estimation. Click OK to obtain the model Xt − 1.0449Xt−1 + 0.2456Xt−2 Zt, {Zt} ∼WN(0, 0.4706), with AICC value 213.55 and 95% confidence bounds φ1 :1.0449 ± 1.0449/5.5295 (0.8559, 1.2339), φ2 : −0.2456 ± 0.2456/1.2997 (−0.4346, −0.0566). Selecting the Yule–Walker method for estimation, we obtain the model ACF -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 5-3 The sample ACF of the lake data in Example 5.1.4. Lag Xt − 1.0538Xt−1 + 0.2668Xt−2 Zt, {Zt} ∼WN(0, 0.4920), 0 10 20 30 40
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Introduction to Time Series and For
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Peter J. Brockwell Richard A. Davis
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viii Preface Since the upgrade to I
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x Contents 2.6. The Wold Decomposit
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xii Contents 8.8.2. Observation-Dri
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xiv Contents D.6. Model Properties
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2 Chapter 1 Introduction Figure 1-1
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4 Chapter 1 Introduction Figure 1-3
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6 Chapter 1 Introduction 3.5% of th
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8 Chapter 1 Introduction Example 1.
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10 Chapter 1 Introduction where mt
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12 Chapter 1 Introduction Figure 1-
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14 Chapter 1 Introduction n/12 6,
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16 Chapter 1 Introduction variable,
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18 Chapter 1 Introduction at once t
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20 Chapter 1 Introduction Example 1
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22 Chapter 1 Introduction ACF -0.2
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28 Chapter 1 Introduction can be co
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30 Chapter 1 Introduction If the op
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32 Chapter 1 Introduction The reest
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36 Chapter 1 Introduction (a) The s
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38 Chapter 1 Introduction It can al
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40 Chapter 1 Introduction Problems
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42 Chapter 1 Introduction 1.8. Let
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44 Chapter 1 Introduction At this p
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46 Chapter 2 Stationary Processes T
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48 Chapter 2 Stationary Processes T
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50 Chapter 2 Stationary Processes i
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52 Chapter 2 Stationary Processes P
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54 Chapter 2 Stationary Processes I
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56 Chapter 2 Stationary Processes A
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58 Chapter 2 Stationary Processes 2
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60 Chapter 2 Stationary Processes w
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62 Chapter 2 Stationary Processes A
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70 Chapter 2 Stationary Processes
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74 Chapter 2 Stationary Processes a
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76 Chapter 2 Stationary Processes t
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80 Chapter 2 Stationary Processes 2
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84 Chapter 3 ARMA Models The proces
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86 Chapter 3 ARMA Models where θ0
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88 Chapter 3 ARMA Models The AR pol
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90 Chapter 3 ARMA Models moving ave
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92 Chapter 3 ARMA Models ACF -0.2 0
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94 Chapter 3 ARMA Models can be fou
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96 Chapter 3 ARMA Models Example 3.
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98 Chapter 3 ARMA Models is entirel
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100 Chapter 3 ARMA Models 3.3 Forec
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102 Chapter 3 ARMA Models and rn
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104 Chapter 3 ARMA Models and 2 E
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106 Chapter 3 ARMA Models where the
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108 Chapter 3 ARMA Models Problems
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110 Chapter 3 ARMA Models 3.10. By
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112 Chapter 4 Spectral Analysis 4.1
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114 Chapter 4 Spectral Analysis (ii
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116 Chapter 4 Spectral Analysis Not
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118 Chapter 4 Spectral Analysis Exa
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120 Chapter 4 Spectral Analysis Fig
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122 Chapter 4 Spectral Analysis Fig
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Page 304: 138 Chapter 5 Modeling and Forecast
Page 330: 5.1 Preliminary Estimation 151 Defi
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Page 338: 5.1 Preliminary Estimation 155 Havi
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Page 358: Figure 5-5 The rescaled residuals a
Page 362: 5.4 Forecasting 5.4 Forecasting 167
Page 366: 5.5 Order Selection 5.5 Order Selec
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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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