Applied Time. Series Analysis. Wayne A. Woodward. Henry L. Gray. Alan C. Elliott. Dallas, Texas, USA

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1 Applied Time Series Analysis Wayne A. Woodward Southern Methodist University Dallas, Texas, USA Henry L. Gray Southern Methodist University Dallas, Texas, USA Alan C. Elliott University of Texas Southwestern Medical Center at Dallas Dallas, Texas, USA CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an Informa business A CHAPMAN & HALL BOOK

2 Contents Preface Acknowledgments xvii xxiii 1. Stationary Time Series Time Series Stationary Time Series Autocovariance and Autocorrelation Functions for Stationary 1.4 Estimation of the Mean, Autocovariance, and Autocorrelation Time Series 7 for Stationary Time Series Estimation of /u, 11 ^ Ergodicity ofx Variance of X Estimation of 7& Estimation of Pk Power Spectrum Estimating the Power Spectrum and Spectral Density for Discrete Time Series Time Series Examples Simulated Data Real Data 40 1.A Appendix 45 Exercises Linear Filters Introduction to Linear Filters Relationship between the Spectra of the Input and Output of a Linear Filter Stationary General Linear Processes Spectrum and Spectral Density for a General Linear Process Wold Decomposition Theorem Filtering Applications Butterworth Filters 63 2.A Appendix 68 Exercises 69 ix

3 x Contents 3. ARMA Time Series Models Moving Average Processes MA(1) Model MA(2) Model Autoregressive Processes Inverting the Operator AR(1) Model AR( p) Model for p > Autocorrelations of an AR(p) Model Linear Difference Equations Spectral Density of an AR(p) Model AR(2) Model Autocorrelations of an AR(2) Model Spectral Density of an AR(2) Stationary/Causal Region of an AR(2) (//-Weights of an AR(2) Model Summary of AR(1) and AR(2) Behavior AR(p) Model AR(1) and AR(2) Building Blocks of an AR(p) Model Factor Tables Ill Mvertibilily/Infinite-Order Autoregressive Processes Two Reasons for Imposing Invertibility Autoregressive-Moving Average Processes Stationarity and Invertibility Conditions for an ARMA(p^) Model Spectral Density of an ARMA( p,q) Model Factor Tables and ARMA(p,(?) Models Autocorrelations of an ARMA(p^) Model (//-Weights of an ARMA(p/(j) Approximating ARMA(p^) Processes Using High-Order AR(p) Models Visualizing Autoregressive Components Seasonal ARMA(p,<?) x (PS,QS)S Models 136 Processes Generating Realizations from ARMA(p,q) MA((j) Model AR(2) Model General Procedure Transformations Memoryless Autoregressive Transformations A Appendix: Proofs of Theorems 147 Transformations 142 Exercises 151

4 Contents xi 4. Other Stationary Time Series Models Stationary Harmonic Models Pure Harmonic Models Harmonic Signal-plus-Noise Models ARMA Approximation to the Harmonic Signal-plus-Noise Model ARCH and GARCH Processes 168 Exercises ARCH Processes The ARCH(l) Model The ARCH(g0) Model The GARCH(p0,(?o) Process AR Processes with ARCH or GARCH Noise Nonstationary Time Series Models Deterministic Signal-plus-Noise Models Trend-Component Models Harmonic Component Models ARIMA(p,d,c/) and ARUMA(p,d,<?) Processes Extended Autocorrelations of an ARUMA(p,d,^) Process Cyclical Models Multiplicative Seasonal ARUMA(p,d,c?) x {PS,DS,QS)S Process Factor Tables for Seasonal Models of the Form (5.17) with s = 4 and s 12 = Random Walk Models Random Walk Random Walk with Drift G-Stationary Models for Data with Time-Varying Frequencies 194 Exercises Forecasting Mean Square Prediction Background Box-Jenkins Forecasting for ARMA(p,q) Models Properties of the Best Forecast r-Weight Form of the Forecast Function Forecasting Based on the Difference Equation Eventual Forecast Function Probability Limits for Forecasts Forecasts Using AR\MA(p,d,q) Models Forecasts Using Multiplicative Seasonal ARUMA Models Forecasts Based on Signal-plus-Noise Models A Appendix 229 Exercises 230

5 xii Contents 7. Parameter Estimation Introduction Preliminary Estimates PreUminary Estimates for AR(p) Models Yule-Walker Estimates Least Squares Estimation Burg Estimates Preliminary Estimates for MA(cj) Models Method-of-Moment Estimation foranma^) MA(q) Estimation Using the Innovations Algorithm Preliminary Estimates for ARMA( p,q) Models Extended Yule-Walker Estimates of the Autoregressive Parameters Tsay-Tiao (TT) Estimates of the Autoregressive Parameters Estimating the Moving Average Parameters Maximum Likelihood Estimation of ARMA( p,q) Parameters Conditional and Unconditional Maximum Likelihood Estimation ML Estimation Using the Innovations Algorithm Backcasting and Estimating Asymptotic Properties of Estimators Autoregressive Case Confidence Intervals: Atitoregressive Case ARMA(p^) Case Confidence Intervals for ARMA( p,q) Parameters Asymptotic Comparisons of Estimators foranma(l) Estimation Examples Using Data ARMA Spectral Estimation ARUMA Spectral Estimation 274 Exercises Model Identification Preliminary Check for White Noise Model Identification for Stationary ARMA Models Model Identification Based on AIC and Related Measures 283

6 B)d Contents xiii 8.3 Model Identification for Nonstationary AR\JMA(p,d,Cj) Models Including a Nonstationary Factor in the Model Identifying Nonstationary Component(s) in a Model Decision between a Stationary or a Nonstationary Model Deriving a Final ARUMA Model More on the Identification of Nonstationary Components A Appendix: Including a Factor (1 in the Model Testing for a Unit Root Including a Seasonal Factor (1 Bs) in the Model 299 Model Identification Based on Pattern Recognition 309 Exercises Model Building Residual Analysis Check Sample Autocorrelations of Residuals versus 95% Limit Lines Ljung-Box Test Other Tests for Randomness Testing Residuals for Normality Stationarity versus Nonstationarity Signal-plus-Noise versus Purely Autocorrelation-Driven Models Cochrane Orcutt, ML, and Frequency Domain Method A Bootstrapping Approach Other Methods for Trend Testing Checking Realization Characteristics Comprehensive Analysis of Time Series Data: A Summary 347 Exercises Vector-Valued (Multivariate) Time Series Multivariate Time Series Basics Stationary Multivariate Time Series Estimating the Mean and Covariance for Stationary Multivariate Processes Estimating^ Estimating i 358

7 [v Contents 10.3 Multivariate (Vector) ARMA Processes Forecasting Using VAR(p) Models Spectrum of a VAR(p) Model Estimating the Coefficients of a VAR(p) Model Yule-Walker Estimation Least Squares and Conditional Maximum Likelihood Estimation Burg-Type Estimation Calculating the Residuals and Estimating Ta VAR(p) Spectral Density Estimation Fitting a VAR(p) Model to Data Model Selection Estimating the Parameters Testing the Residuals for White Noise Nonstationary VARMA Processes Testing for Association between Time Series Testing for Independence of Two Stationary Time Series Testing for Cointegration between Nonstationary Time Series State-Space Models State Equation Observation Equation Goals of State-Space Modeling Kalman Filter Prediction (Forecasting) Filtering Smoothing Using the Kalman Filter ft-step Ahead Predictions Kalman Filter and Missing Data Parameter Estimation Using State-Space Methods to Find Additive Components of a Univariate Autoregressive Realization Revised State-Space Model i/^real Complex A Appendix: Derivation of State-Space Results 393 Exercises Long-Memory Processes Long Memory Fractional Difference and FARMA Processes Gegenbauer and GARMA Processes Gegenbauer Polynomials 410

8 Contents xv Gegenbauer Process GARMA Process fc-factor Gegenbauer and GARMA Processes Calculating Autocovariances Generating Realizations Parameter Estimation and Model Identification Forecasting Based on the fc-factor GARMA Model Modeling Atmospheric C02 Data Using Long-Memory Models 429 Exercises Wavelets Shortcomings of Traditional Spectral Analysis for TVF Data Window-Based Methods That Localize the "Spectrum" in Time Gabor Spectrogram Wigner-Ville Spectrum Wavelet Analysis Fourier Series Background Wavelet Analysis Introduction Fundamental Wavelet Approximation Result Discrete Wavelet Transform for Data Sets of Finite Length Pyramid Algorithm Multiresolution Analysis Wavelet Shrinkage Scalogram: Time-Scale Plot Wavelet Packets Two-Dimensional Wavelets Concluding Remarks on Wavelets A Appendix: Mathematical Preliminaries for This Chapter 473 Exercises G-Stationary 13.1 Generalized-Stationary Processes General Strategy for Analyzing G-Stationary Processes 480 Processes M-Stationary Continuous M-Stationary Process 481 Processes Discrete M-Stationary Process Discrete Euler(p) Model Time Transformation and Sampling 484

9 xvi Contents 13.3 G(A)-Stationary Processes Continuous G(p;A) Model Sampling the Continuous G(A)-Stationary Equally Spaced Sampling from Processes G(p;A) Processes Analyzing TVP Data Using the G(p;A) Model G(p;A) Spectral Density Linear Chirp Processes Models for Generalized Linear Chirps Concluding Remarks A Appendix 512 Exercises 516 References 519 Index 529

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