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Figure 1-5 Population of the U.S.A. at ten-year intervals, 1790–1990. Figure 1-6 Strikes in the U.S.A., 1951–1980. 1.1 Examples of Time Series 5 (Millions) 0 20 40 60 80 120 160 200 240 (thousands) 3.5 4.0 4.5 5.0 5.5 6.0 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 1950 1960 1970 1980
Page 1 and 2: Introduction to Time Series and For
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Page 8: Peter J. Brockwell Richard A. Davis
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Page 16: viii Preface Since the upgrade to I
Page 20: x Contents 2.6. The Wold Decomposit
Page 24: xii Contents 8.8.2. Observation-Dri
Page 28: xiv Contents D.6. Model Properties
Page 32: 2 Chapter 1 Introduction Figure 1-1
Page 36: 4 Chapter 1 Introduction Figure 1-3
Page 42: 1.3 Some Simple Time Series Models
Page 50: Figure 1-8 Population of the U.S.A.
Page 58: 1.4 Stationary Models and the Autoc
Page 74: 1.5 Estimation and Elimination of T
Page 78: 1.5 Estimation and Elimination of T
Page 82: Figure 1-20 Smoothing with a low-pa
Page 86: Figure 1-22 Strike data smoothed by
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1.5 Estimation and Elimination of T
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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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