Long-Memory Time Series

Wilfredo Palma, PhD, is Chairman and Professor of Statistics in the Department of Statistics at Pontificia Universidad Católica de Chile. Dr. Palma has published several refereed articles and has received over a dozen academic honors and awards. His research interests include time series analysis, prediction theory, state space systems, linear models, and econometrics.

"...Palma presents a textbook for a graduate course summarizing the theory and methods developed to deal with long-range-dependent data, and describing some applications to real-life time series." (SciTech Book Reviews, June 2007) "...textbook for a graduate course summarizing the theory and methods developed to deal with long-range-dependent data, and describing some applications to real-life time series.... Problems and bibliographic notes are provided at the end of each chapter." (SciTech Book News, June 2007)"I believe that this text provides an important contribution to the long-memory time series literature. I feel that it largely achieves its aims and could be useful for those instructors wishing to teach a semester-long special topics course.... I strongly recommend this book to anyone interested in long-memory time series. Both researchers and beginners alike will find this text extremely useful." (Journal of the American Statisticial Association, Dec 2008)"Very well-organized catalogue of long-memory time series analysis." (Mathematical Reviews, 2008)"Judging by its contents and scope [the aim of this book] has been largely achieved.... The list of references is selective but quite comprehensive. Each chapter concludes with a 'Problems' section which should be helpful to instructors wishing to use this book as standalone basis for a course in its subject area..." (International Statistical Review, 2007)

• Preface xiiiAcronyms xvii1 Stationary Precedes 11.1 Fundamental concepts 21.1.1 Stationarity 41.1.2 Singularity and Regularity 51.1.3 Wold Decomposition Theorem 51.1.4 Causality 71.1.5 Invertibility 71.1.6 Best Linear Predictor 81.1.7 Szego-Kolmogorov Formula 81.1.8 Ergodicity 91.1.9 Martingales 111.1.10 Cumulants 121.1.11 Fractional Brownian Motion 121.1.12 Wavelets 141.2 Bibliographic Notes 15Problems 162 State Space Systems 212.1 Introduction 222.1.1 Stability 222.1.2 Hankel Operator 222.1.3 Observability 232.1.4 Controllability 232.1.5 Minimality 242.2 Representations of Linear Processes 242.2.1 State Space Form to Wold Decomposition 242.2.2 Wold Decomposition to State Form 252.2.3 Hankel Operator to State Space Form 252.3 Estimation of the State 262.3.1 State Predictor 272.3.2 State Filter 272.3.3 State Smoother 272.3.4 Missing Observation 282.3.5 Steady State System 282.3.6 Prediction of Future Observations 302.4 Extensions 322.5 Bibliographic Notes 32Problems 333 Long-Memory/Processes 393.1 Defining Long Memory 403.1.1 Alternative Definitions 413.1.2 Extensions 433.2 ARFIMA Processes 433.2.1 Stationarity, Causality, and Invertibility 443.2.2 Infinite AR and MA Expansions 463.2.3 Spectral Density 473.2.4 Autocovariance Function 473.2.5 Sample Mean 483.2.6 Partial Autocorrelations 493.2.7 Illustrations 493.2.8 Approximation of Long-Memory Processes 553.3 Fractional Gaussian Noise 563.3.1 Sample Mean 563.4 Technical Lemmas 573.5 Bibliographic Notes 58Problems 594 Estimation Methods 654.1 Maximum-Likelihood Estimation 664.1.1 Cholesky Decomposition Method 664.1.2 Durbin-Levinson Algorithm 664.1.3 Computation of Autocovariances 674.1.4 State Space Approach 694.2 Autoregressive Approximations 714.2.1 Haslett-Raftery Method724.2.2 Beran Approach 734.2.3 A State Space Method 744.3 Moving-Average Approximation 754.4 Whittle Estimation 784.4.1 Other versions 804.4.2 Non-Gaussian Data 804.4.3 Semiparametric Methods 814.5 Other Methods 814.5.1 A Regression Method 824.5.2 Rescale Range Method 834.5.3 Variance Plots 854.5.4 Detrended Fluctuation Analysis 874.5.5 A Wavelet-Based Method 914.6 Numerical Experiments 924.7 Bibliographic Notes 93Problems 945 Asymptotic Theory 975.1 Notation and Definitions 985.2 Theorems 995.2.1 Consistency 995.2.2 Central Limit Theorem 1015.2.3 Efficiency 1045.3 Examples 1045.4 Illustration 1085.5 Technical Lemmas 1095.6 Bibliographic Notes 109Problems 1096 Heteroskedastic Models 1156.1 Introduction 1166.2 ARFIMA-GARCH Model 1176.2.1 Estimation 1196.3 Other Models 1196.3.1 Estimation 1216.4 Stochastic Volatility 1216.4.1 Estimation 1226.5 Numerical Experiments 1226.6 Application 1236.6.1 Model without Leverage 1236.6.2 Model with Leverage 1246.6.3 Model Comparison 1246.7 Bibliographic Notes 125Problems 1267 Transformations 1317.1 Transformation of Gaussian Processes 1327.2 Autocorrelation of Squares 1347.3 Asymptotic behavior 1367.4 Illustrations 1387.5 Bibliographic Notes 142Problems 1438 Bayesian Methods 1478.1 Bayesian Modeling 1488.2 Markov Chain Monte Carlo Methods 1498.2.1 Metropolis-Hastings Algorithm 1498.2.2 Gibbs Sampler 1508.2.3 Overdispersed Distributions 1528.3 Monitoring Convergence 1538.4 A Simulated Example 1558.5 Data Application 1588.6 Bibliographic Notes 162Problems 1629 Prediction 1679.1 One-Step Ahead Predictors 1689.1.1 Infinite Past 1689.1.2 Finite Past 1689.1.3 An Approximate Predictor 1729.2 Multistep Ahead Predictors 1739.2.1 Infinite Past 1739.2.2 Finite Past 1749.3 Heteroskedastic Models 1759.3.1 Prediction of Volatility 1769.4 Illustration 1789.5 Rational Approximations 1809.5.1 Illustration 1829.6 Bibliographic Notes Problems 18410 Regression 187 10.1 Linear Regression Model 18810.1.1 Grenander conditions 18810.2 Properties of the LSE 19110.2.1 Consistency 19210.2.2 Asymptotic Variance 19310.2.3 Asymptotic Normality 19310.3 Properties of the BLUE 19410.3.1 Efficiency of the LSE Relative to the BLUE 19510.4 Estimation of the Mean 19810.4.1 Consistency 19810.4.2 Asymptotic Variance 19910.4.3 Normality 20010.4.4 Relative Efficiency 20010.5 Polynomial Trend 20210.5.1 Consistency 20310.5.2 Asymptotic Variance 20310.5.3 Normality 20410.5.4 Relative Efficiency 20410.6 Harmonic Regression 20510.6.1 Consistency 20510.6.2 Asymptotic Variance 20510.6.3 Normality 20510.6.4 Efficiency 20610.7 Illustration: Air Pollution Data 20710.8 Bibliographic Notes 210Problems 21111 Missing Data 21511.1 Motivation 21611.2 Likelihood Function with Incomplete Data 21711.2.1 Integration 21711.2.2 Maximization 21811.2.3 Calculation of the Likelihood Function 21911.2.4 Kalman Filter with Missing Observations 21911.3 Effects of Missing Values on ML Estimates 22111.3.1 Monte Carlo Experiments 22211.4 Effects of Missing Values on Prediction 22311.5 Illustrations 22711.6 Interpolation of Missing Data 22911.6.1 Bayesian Imputation 23411.6.2 A Simulated Example 23511.7 Bibliographic Notes 239Problems 23912 Seasonality 24512.1 A Long-Memory Seasonal Model 24612.2 Calculation of the Asymptotic Variance 25012.3 Autocovariance Function 25212.4 Monte Carlo Studies 25412.5 Illustration 25812.6 Bibliographic Notes 260Problems 261References 265Topic Index 279Author Index 283