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10.3 Nonlinear Models 349 is a partition of R p , and {Zt} ∼IID(0, 1), then the k difference equations Xt σ (i) Zt + p j1 φ (i) j Xt−j, (Xt−1, ···,Xt−p) ∈ R (i) , i 1, ···,k, (10.3.5) define a threshold AR(p) model. Model identification and parameter estimation for threshold models can be carried out in a manner similar to that for linear models using maximum likelihood and the AIC criterion. 10.3.5 Modeling Volatility For modeling changing volatility as discussed above under deviations from linearity, Engle (1982) introduced the ARCH(p) process {Xt} as a solution of the equations Zt htet, {et} ∼IID N(0, 1), (10.3.6) where ht is the (positive) function of {Zs,s 0 and αj ≥ 0, j 1,...,p. The name ARCH signifies autoregressive conditional heteroscedasticity. ht is the conditional variance of Zt given {Zs,s
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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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10 Chapter 1 Introduction where mt
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22 Chapter 1 Introduction ACF -0.2
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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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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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92 Chapter 3 ARMA Models ACF -0.2 0
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112 Chapter 4 Spectral Analysis 4.1
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138 Chapter 5 Modeling and Forecast
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318 Chapter 9 Forecasting Technique
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Page 734: 10.3 Nonlinear Models 353 process {
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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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