Time Series Analysis

The last decade has brought dramatic changes in the way that researchers analyze economic and financial time series. This book synthesizes these recent advances and makes them accessible to first-year graduate students. James Hamilton provides the first adequate text-book treatments of important innovations such as vector autoregressions, generalized method of moments, the economic and statistical consequences of unit roots, time-varying variances, and nonlinear time series models. In addition, he presents basic tools for analyzing dynamic systems (including linear representations, autocovariance generating functions, spectral analysis, and the Kalman filter) in a way that integrates economic theory with the practical difficulties of analyzing and interpreting real-world data. __Time Series Analysis__ fills an important need for a textbook that integrates economic theory, econometrics, and new results. The book is intended to provide students and researchers with a self-contained survey of time series analysis. It starts from first principles and should be readily accessible to any beginning graduate student, while it is also intended to serve as a reference book for researchers. Contents PREFACE 1 Difference Equations 1.1. First-Order Difference Equations 1.2. pth-Order Difference Equations APPENDIX 1.A. Proofs of Chapter 1 Propositions References 2 Lag Operators 2.1. Introduction 2.2. First-Order Difference Equations 2.3. Second-Order Difference Equations 2.4. pth-Order Difference Equations 2.5. Initial Conditions and Unbounded Sequences References 3 Stationary ARMA Processes 3.1. Expectations, Stationarity, and Ergodicity 3.2. White Noise 3.3. Moving Average Processes 3.4. Autoregressive Processes 3.5. Mixed Autoregressive Moving Average Processes 3.6. The Autocovariance-Generating Function 3.7. Invertibility APPENDIX 3.A. Convergence Results for Infinite-Order Moving Average Processes Exercises References 4 Forecasting 4.1. Principles of Forecasting 4.2. Forecasts Based on an Infinite Numberof Observations 4.3. Forecasts Based on a Finite Numberof Observations 4.4. The Triangular Factorization of a Positive DefiniteSymmetric Matrix 4.5. Updating a Linear Projection 4.6. Optimal Forecasts for Gaussian Processes 4.7. Sums of ARMA Processes 4.8. Wold's Decomposition and the Box-Jenkins Modeling Philosophy APPENDIX 4.A. Parallel Between OLS Regression and Linear Projection APPENDIX 4.B. Triangular Factorization of the Covariance Matrix for an MAA) Process Exercises References 5 Maximum Likelihood Estimation 5.1. Introduction 5.2. The Likelihood Function for a Gaussian AR(1) Process 5.3. The Likelihood Function for a Gaussian AR(p) Process 5.4. The Likelihood Function for a Gaussian MA(1) Process 5.5. The Likelihood Function for a Gaussian MA(q) Process 5.6. The Likelihood Function for a Gaussian ARMA(p, q) Process 5.7. Numerical Optimization 5.8. Statistical Inference with Maximum Likelihood Estimation 5.9. Inequality Constraints APPENDIX 5.A. Proofs of Chapter 5 Propositions Exercises References 6 Spectral Analysis 6.1. The Population Spectrum 6.2. The Sample Periodogram 6.3. Estimating the Population Spectrum 6.4. Uses of Spectral Analysis APPENDIX 6.A. Proofs of Chapter 6 Propositions Exercises References 7 Asymptotic Distribution Theory 7.1. Review of Asymptotic Distribution Theory 7.2. Limit Theorems for Serially Dependent Observations APPENDIX 7. A. Proofs of Chapter 7 Propositions Exercises References 8 Linear Regression Models 8.1. Review of Ordinary Least Squares with Deterministic Regressors and i.i.d. Gaussian Disturbances 8.2. Ordinary Least Squares Under More General Conditions 8.3. Generalized Least Squares APPENDIX 8.A. Proofs of Chapter 8 Propositions Exercises References 9 Linear Systems of Simultaneous Equations 9.1. Simultaneous Equations Bias 9.2. Instrumental Variables and Two-Stage Least Squares 9.3. Identification 9.4. Full-Information Maximum Likelihood Estimation 9.5. Estimation Based on the Reduced Form 9.6. Overview of Simultaneous Equations Bias APPENDIX 9.A. Proofs of Chapter 9 Proposition Exercises References 10 Covariance-Stationary Vector Processes 10.1. Introduction to Vector Autoregressions 10.2. Autocovariances and Convergence Results for Vector Processes 10.3. The Autocovariance-Generating Function for Vector Processes 10.4. The Spectrum for Vector Processes 10.5. The Sample Mean of a Vector Process APPENDIX 10.A. Proofs of Chapter 10 Propositions Exercises References 11 Vector Autoregressions 11.1. Maximum Likelihood Estimation and Hypothesis Testing for an Unrestricted Vector Autoregression 11.2. Bivariate Granger Causality Tests 11.3. Maximum Likelihood Estimation of Restricted Vector Autoregressions 11.4. The Impulse-Response Function 11.5. Variance Decomposition 11.6. Vector Autoregressions and Structural Econometric Models 11.7. Standard Errors for Impulse-Response Functions APPENDIX 11. A. Proofs of Chapter 11 Propositions APPENDIX ll.B. Calculation of Analytic Derivatives Exercises References 12 Bayesian Analysis 12.1. Introduction to Bayesian Analysis 12.2. Bayesian Analysis of Vector Autoregressions 12.3. Numerical Bayesian Methods APPENDIX 12. A. Proofs of Chapter 12 Propositions Exercises References 13 The Kalman Filter 13.1. The State-Space Representation of a Dynamic System 13.2. Derivation of the Kalman Filter 13.3. Forecasts Based on the State-Space Representation 13.4. Maximum Likelihood Estimationof Parameters 13.5. The Steady-State Kalman Filter 13.6. Smoothing 13.7. Statistical Inference with the Kalman Filter 13.8. Time-Varying Parameters APPENDIX 13.A. Proofs of Chapter 13 Propositions Exercises References 14 Generalized Method of Moments 14.1. Estimation by the Generalized Method of Moments 14.2. Examples 14.3. Extensions 14.4. GMM and Maximum Likelihood Estimation APPENDIX 14.A. Proofs of Chapter 14 Propositions Exercises References 15 Models of Nonstationary Time Series 15.1. Introduction 15.2. Why Linear Time Trends and Unit Roots? 15.3. Comparison of Trend-Stationary and Unit Root Processes 15.4. The Meaning of Tests for Unit Roots 15.5. Other Approaches to Trended Time Series APPENDIX 15.A. Derivation of Selected Equations for Chapter 15 References 16 Processes with Deterministic Time Trends 16.1. Asymptotic Distribution of OLS Estimates of the Simple Time Trend Model 16.2. Hypothesis Testing for the Simple Time TrendModel 16.3. Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend APPENDIX 16. A. Derivation of Selected Equationsfor Chapter 16 Exercises References 17 Univariate Processes with Unit Roots 17.1. Introduction 17.2. Browain Motion 17.3. The Functional Central Limit Theorem 17.4. Asymptotic Properties of a First-Order Autoregression when the True Coefficient Is Unity 17.5. Asymptotic Results for Unit Root Processes with General Serial Correlation 17.6. Phillips-Perron Tests for Unit Roots 17.7. Asymptotic Properties of a pth-Order Autoregression and the Augmented Dickey-Fuller Tests for Unit Roots 17.8. Other Approaches to Testing for Unit Roots 17.9. Bayesian Analysis and Unit Roots APPENDIX 17.A. Proofs of Chapter 17 Propositions Exercises References 18 Unit Roots in Multivariate Time Series 18.1. Asymptotic Results for Nonstationary Vector Processes 18.2. Vector Autoregressions Containing Unit Roots 18.3. Spurious Regressions APPENDIX 18.A. Proofs of Chapter 18 Propositions Exercises References 19 Cointegration 19.1. Introduction 19.2. Testing the Null Hypothesis of No Cointegration 19.3. Testing Hypotheses About the Cointegrating Vector APPENDIX 19. A. Proofs of Chapter 19 Propositions Exercises References 20 Full-Information Maximum Likelihood Analysis of Cointegrated Systems 20.1. Canonical Correlation 20.2. Maximum Likelihood Estimation 20.3. Hypothesis Testing 20.4. Overview of Unit Roots—To Difference or Not to Difference? APPENDIX 20. A. Proofs of Chapter 20 Propositions Exercises References 21 Time Series Models of Heteroskedasticity 21.1 Autoregressive Conditional Heteroskedasticity (ARCH) 21.2. Extensions APPENDIX 21.A. Derivation of Selected Equationsfor Chapter 21 References 22 Modeling Time Series with Changesin Regime 22.1. Introduction 22.2. Markov Chains 22.3. Statistical Analysis of i.i.d. Mixture Distributions 22.4. Time Series Models of Changes in Regime APPENDIX 22. A. Derivation of Selected Equationsfor Chapter 22 Exercises References A Mathematical Review A.1. Trigonometry A.2. Complex Numbers A.3. Calculus A.4. Matrix Algebra A.5. Probability and Statistics References B Statistical Tables C Answers to Selected Exercises D Greek Letters and Mathematical Symbols Used in the Text AUTHOR INDEX SUBJECT INDEX The Last Decade Has Brought Dramatic Changes In The Way That Researchers Analyze Time Series Data. This Much-needed Book Synthesizes All Of The Major Recent Advances And Develops A Single, Coherent Presentation Of The Current State Of The Art Of This Increasingly Important Field. James Hamilton Provides For The First Time A Thorough And Detailed Textbook Account Of Important Innovations Such As Vector Autoregressions, Estimation By Generalized Method Of Moments, The Economic And Statistical Consequences Of Unit Roots, Time-varying Variances, And Nonlinear Time Series Models. In Addition, Hamilton Presents Traditional Tools For Analyzing Dynamic Systems, Including Linear Representations, Autocovariance, Generating Functions, Spectral Analysis, And The Kalman Filter, Illustrating Their Usefulness Both For Economic Theory And For Studying And Interpreting Real-world Data. This Book Is Intended To Provide Students, Researchers, And Forecasters With A Definitive, Self-contained Survey Of Dynamic Systems, Econometrics, And Time Series Analysis. Starting From First Principles, Hamilton's Lucid Presentation Makes Both Old And New Developments Accessible To First-year Graduate Students And Nonspecialists. Moreover, The Work's Thoroughness And Depth Of Coverage Will Make Time Series Analysis An Invaluable Reference For Researchers At The Frontiers Of The Field. Hamilton Achieves These Dual Objectives By Including Numerous Examples That Illustrate Exactly How The Theoretical Results Are Used And Applied In Practice, While Relegating Many Details To Mathematical Appendixes At The End Of Chapters. As An Intellectual Roadmap Of The Field For Students And Researchers Alike, This Volume Promises To Be The Authoritative Guide For Years To Come.--jacket. Difference Equations -- Lag Operators -- Stationary Arma Processes -- Forecasting -- Maximum Likelihood Estimation -- Spectral Analysis -- Asymptotic Distribution Theory -- Linear Regression Models -- Linear Systems Of Simultaneous Equations -- Covariance-stationary Vector Processes -- Vector Autoregressions -- Bayesian Analysis -- The Kalman Filter -- Generalized Method Of Moments -- Models Of Nonstationary Time Series -- Processes With Deterministic Time Trends -- Univariate Processes With Unit Roots -- Unit Roots In Multivariate Time Series -- Cointegration -- Full-information Maximum Likelihood Analysis Of Cointegrated Systems -- Time Series Models Of Heteroskedasticity -- Modeling Time Series With Changes In Regime. James D. Hamilton. Includes Bibliographical References And Indexes. The last decade has brought dramatic changes in the way that researchers analyze economic and financial time series. This book synthesizes these recent advances and makes them accessible to first-year graduate students. James Hamilton provides the first adequate text-book treatments of important innovations such as vector autoregressions, generalized method of moments, the economic and statistical consequences of unit roots, time-varying variances, and nonlinear time series models. In addition, he presents basic tools for analyzing dynamic systems (including linear representations, autocovariance generating functions, spectral analysis, and the Kalman filter) in a way that integrates economic theory with the practical difficulties of analyzing and interpreting real-world data. Time Series Analysis fills an important need for a textbook that integrates economic theory, econometrics, and new results. The book is intended to provide students and researchers with a self-contained survey of time series analysis. It starts from first principles and should be readily accessible to any beginning graduate student, while it is also intended to serve as a reference book for researchers. source: https://press.princeton.edu/titles/5386.html The last decade has brought dramatic changes in the way that researchers analyze economic and financial time series. This book synthesizes these recent advances and makes them accessible to first-year graduate students. James Hamilton provides the first adequate text-book treatments of important innovations such as vector autoregressions, generalized method of moments, the economic and statistical consequences of unit roots, time-varying variances, and nonlinear time series models. In addition, he presents basic tools for analyzing dynamic systems (including linear representations, autocovariance generating functions, spectral analysis, and the Kalman filter) in a way that integrates economic theory with the practical difficulties of analyzing and interpreting real-world data. Time Series Analysis fills an important need for a textbook that integrates economic theory, econometrics, and new results. The book is intended to provide students and researchers with a self-contained survey of time series analysis. It starts from first principles and should be readily accessible to any beginning graduate student, while it is also intended to serve as a reference book for researchers.-- "Journal of Economics" An authoritative, self-contained overview of time series analysis for students and researchers The past decade has brought dramatic changes in the way that researchers analyze economic and financial time series. This textbook synthesizes these advances and makes them accessible to first-year graduate students. James Hamilton provides comprehensive treatments of important innovations such as vector autoregressions, generalized method of moments, the economic and statistical consequences of unit roots, time-varying variances, and nonlinear time series models. In addition, he presents basic tools for analyzing dynamic systems—including linear representations, autocovariance generating functions, spectral analysis, and the Kalman filter—in a way that integrates economic theory with the practical difficulties of analyzing and interpreting real-world data. Time Series Analysis fills an important need for a textbook that integrates economic theory, econometrics, and new results. This invaluable book starts from first principles and should be readily accessible to any beginning graduate student, while it is also intended to serve as a reference book for researchers.