TY  - BOOK
AU  - Woodward,Wayne A
AU  - Gray,Henry L
AU  - Elliott,Alan C.
TI  - Applied time series analysis
T2  - Statistics: textbooks and monographs
SN  - 9781439818374
AV  - QA280 .W66 2012
PY  - 2012///
CY  - Boca Raton
PB  - Chapman and Hall/CRC
KW  - Time-series analysis
N1  - 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
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