Time Series: Theory and Methods

Transcription

1 Peter J. Brockwell Richard A. Davis Time Series: Theory and Methods Second Edition With 124 Illustrations Springer

2 Contents Preface to the Second Edition Preface to the First Edition vn ix CHAPTER 1 Stationary Time Series Examples of Time Series Stochastic Processes Stationarity and Strict Stationarity The Estimation and Elimination of Trend and Seasonal Components The Autocovariance Function of a Stationary Process The Multivariate Normal Distribution * Applications of Kolmogorov's Theorem 37 Problems 39 CHAPTER 2 Hubert Spaces Inner-Product Spaces and Their Properties Hubert Spaces The Projection Theorem Orthonormal Sets Projection in W Linear Regression and the General Linear Model Mean Square Convergence, Conditional Expectation and Best Linear Prediction in L 2 (Q, &, P) Fourier Series Hubert Space Isomorphisms * TheCompletenessofL 2 (Q,,^",P) * Complementary Results for Fourier Series 69 Problems 73

3 XIV Contents CHAPTER 3 Stationary ARMA Processes Causal and Invertible ARMA Processes Moving Average Processes of Infinite Order Computing the Autocovariance Function of an ARMA(p, q) Process The Partial Autocorrelation Function The Autocovariance Generating Function * Homogeneous Linear Difference Equations with Constant Coefficients 105 Problems 110 CHAPTER 4 The Spectral Representation of a Stationary Process Complex-Valued Stationary Time Series The Spectral Distribution of a Linear Combination of Sinusoids Herglotz's Theorem Spectral Densities and ARMA Processes * Circulants and Their Eigenvalues * Orthogonal Increment Processes on [ n, 7t] * Integration with Respect to an Orthogonal Increment Process * The Spectral Representation * Inversion Formulae * Time-Invariant Linear Filters * Properties of the Fourier Approximation h to 7 (V w] 157 Problems 159 CHAPTER 5 Prediction of Stationary Processes The Prediction Equations in the Time Domain Recursive Methods for Computing Best Linear Predictors Recursive Prediction of an ARMA(p, q) Process Prediction of a Stationary Gaussian Process; Prediction Bounds Prediction of a Causal Invertible ARMA Process in Terms of X p oo <j <n * Prediction in the Frequency Domain * The Wold Decomposition * Kolmogorov's Formula 191 Problems 192 CHAPTER 6* Asymptotic Theory Convergence in Probability Convergence in r lh Mean, r > Convergence in Distribution Central Limit Theorems and Related Results 209 Problems 215

4 Contents xv CHAPTER 7 Estimation of the Mean and the Autocovariance Function Estimation of n Estimation of y ( ) and p( ) * Derivation of the Asymptotic Distributions 225 Problems 236 CHAPTER 8 Estimation for ARMA Models The Yule-Walker Equations and Parameter Estimation for Autoregressive Processes Preliminary Estimation for Autoregressive Processes Using the Durbin-Levinson Algorithm Preliminary Estimation for Moving Average Processes Using the Innovations Algorithm Preliminary Estimation for ARMA(p, q) Processes Remarks on Asymptotic Efficiency Recursive Calculation of the Likelihood of an Arbitrary Zero-Mean Gaussian Process Maximum Likelihood and Least Squares Estimation for ARMA Processes Asymptotic Properties of the Maximum Likelihood Estimators Confidence Intervals for the Parameters of a Causal Invertible ARMA Process * Asymptotic Behavior of the Yule-Walker Estimates * Asymptotic Normality of Parameter Estimators 265 Problems 269 CHAPTER 9 Model Building and Forecasting with ARIMA Processes ARIMA Models for Non-Stationary Time Series Identification Techniques Order Selection Diagnostic Checking Forecasting ARIMA Models Seasonal ARIMA Models 320 Problems 326 CHAPTER 10 Inference for the Spectrum of a Stattonary Process The Periodogram TestingforthePresenceofHidden Periodicities Asymptotic Properties of the Periodogram Smoothing the Periodogram Confidence Intervals for the Spectrum Autoregressive, Maximum Entropy, Moving Average and Maximum Likelihood ARMA Spectral Estimators The Fast Fourier Transform (FFT) Algorithm 373

5 XVI Contents 10.8* Derivation of the Asymptotic Behavior of the Maximum Likelihood and Least Squares Estimators of the Coefficients of an ARMA Process 375 Problems 396 CHAPTER 11 Multivariate Time Series Second Order Properties of Multivariate Time Series Estimation of the Mean and Covariance Function Multivariate ARMA Processes Best Linear Predictors of Second Order Random Vectors Estimation for Multivariate ARMA Processes The Cross Spectrum Estimating the Cross Spectrum * The Spectral Representation of a Multivariate Stationary Time Series 454 Problems 459 CHAPTER 12 State-Space Models and the Kaiman Recursions State-Space Models The Kaiman Recursions State-Space Models with Missing Observations Controllability and Observability Recursive Bayesian State Estimation 498 Problems 501 CHAPTER 13 Further Topics Transfer Function Modelling Long Memory Processes Linear Processes with Infinite Variance Threshold Models 545 Problems 552 Appendix: Data Sets 555 Bibliography 561 Index 567