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302 Chapter 8 State-Space Models is said to belong to an exponential family (in natural parameterization) if p(yt|xt) exp{ytxt − b(xt) + c(yt)}, (8.8.31) where b(·) is a twice continuously differentiable function and c(yt) does not depend on xt. This family includes the normal, exponential, gamma, Poisson, binomial, and many other distributions frequently encountered in statistics. Detailed properties of the exponential family can be found in Barndorff-Nielsen (1978), and an excellent treatment of its use in the analysis of linear models is given by McCullagh and Nelder (1989). We shall need only the following important facts: e b(xt ) exp{ytxt + c(yt)} ν(dyt), (8.8.32) b ′ (xt) E(Yt|xt), (8.8.33) b ′′ (xt) Var(Yt|xt) : y 2 t p(yx|xt) ν(dyt) − b ′ (xt) 2 , (8.8.34) where integration with respect to ν(dyt) means integration with respect to dyt in the continuous case and summation over all values of yt in the discrete case. Proof of (8.8.32)–(8.8.34) The first relation is simply the statement that p(yt|xt) integrates to 1. The second relation is established by differentiating both sides of (8.8.32) with respect to xt and then multiplying through by e −b(xt ) (for justification of the differentiation under the integral sign see Barndorff-Nielson (1978)). The last relation is obtained by differentiating (8.8.32) twice with respect to xt and simplifying. Example 8.8.6 The Poisson case If the observation Yt, givenXt xt, has a Poisson distribution of the form (8.8.21), then p(yt|xt) exp ytxt − e xt − ln yt! , yt 0, 1,..., (8.8.35) which has the form (8.8.31) with b(xt) e xt and c(yt) −ln yt!. From (8.8.33) we easily find that E(Yt|xt) b ′ (xt) e xt . This parameterization is slightly different from the one used in Examples 8.8.2 and 8.8.5, where the conditional mean of Yt given xt was πxt and not e xt . For this observation equation, define the family of densities f(x; α, λ) exp{αx − λb(x) + A(α, λ)}, −∞ 0 are parameters and A(α, λ) −ln Ɣ(α) + α ln λ. Now consider state densities of the form p(xt+1|y (t) ) f(xt+1; αt+1|t,λt+1|t), (8.8.37)
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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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62 Chapter 2 Stationary Processes A
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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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124 Chapter 4 Spectral Analysis The
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130 Chapter 4 Spectral Analysis Sin
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136 Chapter 4 Spectral Analysis 4.9
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138 Chapter 5 Modeling and Forecast
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180 Chapter 6 Nonstationary and Sea
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224 Chapter 7 Multivariate Time Ser
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260 Chapter 8 State-Space Models li
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262 Chapter 8 State-Space Models In
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264 Chapter 8 State-Space Models Fi
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268 Chapter 8 State-Space Models wh
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272 Chapter 8 State-Space Models De
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Page 634: 8.8 Generalized State-Space Models
Page 642: Figure 8-8 Goals scored by England
Page 650: Problems Problems 311 (see Problem
Page 654: Problems 313 can be expressed as Y
Page 658: Problems 315 and p(x2|y1) g(x2; y1
Page 662: 9 Forecasting Techniques 9.1 The AR
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Page 674: 9.2 The Holt-Winters Algorithm 323
Page 678: 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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