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Contents xiii B. Statistical Complements 383 B.1. Least Squares Estimation 383 B.1.1. The Gauss-Markov Theorem 385 B.1.2. Generalized Least Squares 386 B.2. Maximum Likelihood Estimation 386 B.2.1. Properties of Maximum Likelihood Estimators 387 B.3. Confidence Intervals 388 B.3.1. Large-Sample Confidence Regions 388 B.4. Hypothesis Testing 389 B.4.1. Error Probabilities 390 B.4.2. Large-Sample Tests Based on Confidence Regions 390 C. Mean Square Convergence 393 C.1. The Cauchy Criterion 393 D. An ITSM Tutorial 395 D.1. Getting Started 396 D.1.1. Running ITSM 396 D.2. Preparing Your Data for Modeling 396 D.2.1. Entering Data 397 D.2.2. Information 397 D.2.3. Filing Data 397 D.2.4. Plotting Data 398 D.2.5. Transforming Data 398 D.3. Finding a Model for Your Data 403 D.3.1. Autofit 403 D.3.2. The Sample ACF and PACF 403 D.3.3. Entering a Model 404 D.3.4. Preliminary Estimation 406 D.3.5. The AICC Statistic 408 D.3.6. Changing Your Model 408 D.3.7. Maximum Likelihood Estimation 409 D.3.8. Optimization Results 410 D.4. Testing Your Model 411 D.4.1. Plotting the Residuals 412 D.4.2. ACF/PACF of the Residuals 412 D.4.3. Testing for Randomness of the Residuals 414 D.5. Prediction 415 D.5.1. Forecast Criteria 415 D.5.2. Forecast Results 415
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
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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
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Page 58: 1.4 Stationary Models and the Autoc
Page 74: 1.5 Estimation and Elimination of T
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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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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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