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116 Chapter 4 Spectral Analysis Note that since F(π) γ(0) Var(X1), the ACF of {Xt} has spectral representation ρ(h) e ihλ dG(λ). (−π,π] The function F in (4.1.7) is called the spectral distribution function of γ(·).IfF(λ) can be expressed as F(λ) λ f(y)dyfor all λ ∈ [−π, π], then f is the spectral −π density function and the time series is said to have a continuous spectrum. IfF is a discrete distribution (i.e., if G is a discrete probability distribution), then the time series is said to have a discrete spectrum. The time series (4.1.6) has a discrete spectrum. Example 4.1.2 Linear combination of sinusoids Consider now the process obtained by adding uncorrelated processes of the type defined in (4.1.6), i.e., k Xt (Aj cos(ωjt) + Bj sin(ωjt)), 0
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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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Page 252: 112 Chapter 4 Spectral Analysis 4.1
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Page 262: Figure 4-1 A sample path of size 10
Page 266: 4.1 Spectral Densities 119 where {Z
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Page 274: 4.2 The Periodogram 123 Now e1,...,
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Page 282: 4.3 Time-Invariant Linear Filters 1
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Page 290: Figure 4-12 The transfer function D
Page 294: 4.4 The Spectral Density of an ARMA
Page 298: Problems 135 4.5. If {Xt} and {Yt}
Page 302: 5 Modeling and Forecasting with ARM
Page 306: 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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