Applied time series analysis / Wayne A Woodward, Henry L. Gray, and Alan C. Elliott

Includes bibliographical references and index

Contents Preface Acknowledgments 1. Stationary Time Series 1.1. Time Series 1.2. Stationary Time Series 1.3. Autocovariance and Autocorrelation Functions for Stationary Time Series 1.4. Estimation of the Mean, Autocovariance, and Autocorrelation for Stationary Time Series 1.4.1. Estimation of μ 1.4.1.1. Ergodicity of X 1.4.1.2. Variance of X 1.4.2. Estimation of γk 1.4.3. Estimation of ρk 1.5. Power Spectrum 1.6. Estimating the Power Spectrum and Spectral Density for Discrete Time Series 1.7. Time Series Examples 1.7.1. Simulated Data 1.7.2. Real Data 1.A. Appendix Exercises 2. Linear Filters 2.1. Introduction to Linear Filters 2.1.1. Relationship between the Spectra of the Input and Output of a Linear Filter 2.2. Stationary General Linear Processes 2.2.1. Spectrum and Spectral Density for a General Linear Process 2.3. Wold Decomposition Theorem 2.4. Filtering Applications 2.4.1. Butterworth Filters 2.A. Appendix Exercises 3. ARMA Time Series Models 3.1. Moving Average Processes 3.1.1. MA(1) Model 3.1.2. MA(2) Model 3.2. Autoregressive Processes 3.2.1. Inverting the Operator 3.2.2. AR(1) Model 3.2.3. AR(p) Model for p [≥] 1 3.2.4. Autocorrelations of an AR(p) Model 3.2.5. Linear Difference Equations 3.2.6. Spectral Density of an AR(p) Model 3.2.7. AR(2) Model 3.2.7.1. Autocorrelations of an AR(2) Model 3.2.7.2. Spectral Density of an AR(2) 3.2.7.3. Stationary/Causal Region of an AR(2) 3.2.7.4. ψ-Weights of an AR(2) Model 3.2.8. Summary of AR(1) and AR(2) Behavior 3.2.9. AR(p) Model 3.2.10. AR(1) and AR(2) Building Blocks of an AR(p) Model 3.2.11. Factor Tables 3.2.12. Invertibility/Infinite-Order Autoregressive Processes 3.2.13. Two Reasons for Imposing Invertibility 3.3. Autoregressive-Moving Average Processes 3.3.1. Stationarity and Invertibility Conditions for an ARMA(p,q) Model 3.3.2. Spectral Density of an ARMA(p,q) Model 3.3.3. Factor Tables and ARMA(p,q) Models 3.3.4. Autocorrelations of an ARMA(p,q) Model 3.3.5. ψ-Weights of an ARMA(p,q) 3.3.6. Approximating ARMA(p,q) Processes Using High-Order AR(p) Models 3.4. Visualizing Autoregressive Components 3.5. Seasonal ARMA(p,q) x (PsrQs)s Models 3.6. Generating Realizations from ARMA(p,q) Processes 3.6.1. MA(q) Model 3.6.2. AR(2) Model 3.6.3. General Procedure 3.7. Transformations 3.7.1. Memoryless Transformations 3.7.2. Autoregressive Transformations 3.A. Appendix: Proofs of Theorems Exercises 4. Other Stationary Time Series Models 4.1. Stationary Harmonic Models 4.1.1. Pure Harmonic Models 4.1.2. Harmonic Signal-plus-Noise Models 4.1.3. ARMA Approximation to the Harmonic Signal-plus-Noise Model 4.2. ARCH and GARCH Processes 4.2.1. ARCH Processes 4.2.1.1. The ARCH(1) Model 4.2.1.2. The ARCH(90) Model 4.2.2. The GARCH(po,qo) Process 4.2.3. AR Processes with ARCH or GARCH Noise Exercises 5. Nonstationary Time Series Models 5.1. Deterministic Signal-plus-Noise Models 5.1.1. Trend-Component Models 5.1.2. Harmonic Component Models 5.2. ARIMA(p,d,q) and ARUMA(p,d,q) Processes 5.2.1. Extended Autocorrelations of an ARUMA(p,d,q) Process 5.2.2. Cyclical Models 5.3. Multiplicative Seasonal ARUMA(p,d,q) x (PsrDsrQs)s Process 5.3.1. Factor Tables for Seasonal Models of the Form (5.17) with s = 4 and s = 12 5.4. Random Walk Models 5.4.1. Random Walk 5.4.2. Random Walk with Drift 5.5. G-Stationary Models for Data with Time-Varying Frequencies Exercises 6. Forecasting 6.1. Mean Square Prediction Background 6.2. Box-Jenkins Forecasting for ARMA(p,q) Models 6.3. Properties of the Best Forecast Zto(l) 6.4. π-Weight Form of the Forecast Function 6.5. Forecasting Based on the Difference Equation 6.6. Eventual Forecast Function 6.7. Probability Limits for Forecasts 6.8. Forecasts Using ARUMA(p,d,q) Models 6.9. Forecasts Using Multiplicative Seasonal ARUMA Models 6.10. Forecasts Based on Signal-plus-Noise Models 6.A. Appendix Exercises 7. Parameter Estimation 7.1. Introduction 7.2. Preliminary Estimates 7.2.1. Preliminary Estimates for AR(p) Models 7.2.1.1. Yule-Walker Estimates 7.2.1.2. Least Squares Estimation 7.2.1.3. Burg Estimates 7.2.2. Preliminary Estimates for MA(q) Models 7.2.2.1. Method-of-Moment Estimation for an MA(q) 7.2.2.2. MA(q) Estimation Using the Innovations Algorithm 7.2.3. Preliminary Estimates for ARMA(p,q) Models 7.2.3.1. Extended Yule-Walker Estimates of the Autoregressive Parameters 7.2.3.2. Tsay-Tiao (TT) Estimates of the Autoregressive Parameters 7.2.3.3. Estimating the Moving Average Parameters 7.3. Maximum Likelihood Estimation of ARMA(p,q) Parameters 7.3.1. Conditional and Unconditional Maximum Likelihood Estimation 7.3.2. ML Estimation Using the Innovations Algorithm 7.4. Backcasting and Estimating σ2a 7.5. Asymptotic Properties of Estimators 7.5.1. Autoregressive Case 7.5.1.1. Confidence Intervals: Autoregressive Case 7.5.2. ARMA(p,q) Case 7.5.2.1. Confidence Intervals for ARMA(p,q) Parameters 7.5.3. Asymptotic Comparisons of Estimators for an MA(1) 7.6. Estimation Examples Using Data 7.7. ARMA Spectral Estimation 7.8. ARUMA Spectral Estimation Exercises 8. Model Identification 8.1. Preliminary Check for White Noise 8.2. Model Identification for Stationary ARMA Models 8.2.1. Model Identification Based on AIC and Related Measures 8.3. Model Identification for Nonstationary ARUMA(p,d,q) Models 8.3.1. Including a Nonstationary Factor in the Model 8.3.2. Identifying Nonstationary Component(s) in a Model 8.3.3. Decision between a Stationary or a Nonstationary Model 8.3.4. Deriving a Final ARUMA Model 8.3.5. More on the Identification of Nonstationary Components 8.3.5.1. Including a Factor (1 - B)d in the Model 8.3.5.2. Testing for a Unit Root 8.3.5.3. Including a Seasonal Factor (1 - Bs) in the Model 8.A. Appendix: Model Identification Based on Pattern Recognition Exercises 9. Model Building 9.1. Residual Analysis 9.1.1. Check Sample Autocorrelations of Residuals versus 95% Limit Lines 9.1.2. Ljung-Box Test 9.1.3. Other Tests for Randomness 9.1.4. Testing Residuals for Normality 9.2. Stationarity versus Nonstationarity 9.3. Signal-plus-Noise versus Purely Autocorrelation-Driven Models 9.3.1. Cochrane Orcutt, ML, and Frequency Domain Method 9.3.2. A Bootstrapping Approach 9.3.3. Other Methods for Trend Testing 9.4. Checking Realization Characteristics 9.5. Comprehensive Analysis of Time Series Data: A Summary Exercises 10. Vector-Valued (Multivariate) Time Series 10.1. Multivariate Time Series Basics 10.2. Stationary Multivariate Time Series 10.2.1. Estimating the Mean and Covariance for Stationary Multivariate Processes 10.2.1.1. Estimating μ 10.2.1.2. Estimating π(k) 10.3. Multivariate (Vector) ARMA Processes 10.3.1. Forecasting Using VAR(p) Models 10.3.2. Spectrum of a VAR(p) Model 10.3.3. Estimating the Coefficients of a VAR(p) Model 10.3.3.1. Yule-Walker Estimation 10.3.3.2. Least Squares and Conditional Maximum Likelihood Estimation 10.3.3.3. Burg-Type Estimation 10.3.4. Calculating the Residuals and Estimating πa 10.3.5. VAR(p) Spectral Density Estimation 10.3.6. Fitting a VAR(p) Model to Data 10.3.6.1. Model Selection 10.3.6.2. Estimating the Parameters 10.3.6.3. Testing the Residuals for White Noise 10.4. Nonstationary VARMA Processes 10.5. Testing for Association between Time Series 10.5.1. Testing for Independence of Two Stationary Time Series 10.5.2. Testing for Cointegration between Nonstationary Time Series 10.6. State-Space Models 10.6.1. State Equation 10.6.2. Observation Equation 10.6.3. Goals of State-Space Modeling 10.6.4. Kalman Filter 10.6.4.1. Prediction (Forecasting) 10.6.4.2. Filtering 10.6.4.3. Smoothing Using the Kalman

Filter 10.6.4.4. H-Step Ahead Predictions 10.6.5. Kalman Filter and Missing Data 10.6.6. Parameter Estimation 10.6.7. Using State-Space Methods to Find Additive Components of a Univariate Autoregressive Realization 10.6.7.1. Revised State-Space Model 10.6.7.2. ψ Real 10.6.7.3. ψ Complex 10.A. Appendix: Derivation of State-Space Results Exercises 11. Long-Memory Processes 11.1. Long Memory 11.2. Fractional Difference and FARMA Processes 11.3. Gegenbauer and GARMA Processes 11.3.1. Gegenbauer Polynomials 11.3.2. Gegenbauer Process 11.3.3. GARMA Process 11.4. K-Factor Gegenbauer And Garma Processes 11.4.1. Calculating Autocovariances 11.4.2. Generating Realizations 11.5. Parameter Estimation and Model Identification 11.6. Forecasting Based on the k-Factor GARMA Model 11.7. Modeling Atmospheric CO2 Data Using Long-Memory Models Exercises 12. Wavelets 12.1. Shortcomings of Traditional Spectral Analysis for TVF Data 12.2. Window-Based Methods That Localize the "Spectrum" in Time 12.2.1. Gabor Spectrogram 12.2.2. Wigner-Ville Spectrum 12.3. Wavelet Analysis 12.3.1. Fourier Series Background 12.3.2. Wavelet Analysis Introduction 12.3.3. Fundamental Wavelet Approximation Result 12.3.4. Discrete Wavelet Transform for Data Sets of Finite Length 12.3.5. Pyramid Algorithm 12.3.6. Multiresolution Analysis 12.3.7. Wavelet Shrinkage 12.3.8. Scalogram: Time-Scale Plot 12.3.9. Wavelet Packets 12.3.10. Two-Dimensional Wavelets 12.5. Concluding Remarks on Wavelets 12.A. Appendix: Mathematical Preliminaries for This Chapter Exercises 13. G-Stationary Processes 13.1. Generalized-Stationary Processes 13.1.1. General Strategy for Analyzing G-Stationary Processes 13.2. M-Stationary Processes 13.2.1. Continuous M-Stationary Process 13.2.2. Discrete M-Stationary Process 13.2.3. Discrete Euler(p) Model 13.2.4. Time Transformation and Sampling 13.3. G(λ)-Stationary Processes 13.3.1. Continuous G(p;λ) Model 13.3.2. Sampling the Continuous G(λ)-Stationary Processes 13.3.2.1. Equally Spaced Sampling from G(p;λ) Processes 13.3.3. Analyzing TVF Data Using the G(p;λ) Model 13.3.3.1. G(p;λ) Spectral Density 13.4. Linear Chirp Processes 13.4.1. Models for Generalized Linear Chirps 13.5. Concluding Remarks 13.A. Appendix Exercises References Index