Time Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

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1 Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY &

2 Contents PREFACE xiii Difference Equations First-Order Difference Equations 1 /?th-order Difference Equations 7 APPENDIX l.a. Proofs of Chapter 1 Propositions 21 References Lag Operators Introduction 25 First-Order Difference Equations 27 Second-Order Difference Equations 29 pth-order Difference Equations 33 Initial Conditions and Unbounded Sequences 36 References Stationary ARMA Processes 3.1. Expectations, Stationarity, and Ergodicity White Noise Moving Average Processes Autoregressive Processes Mixed Autoregressive Moving Average Processes 59 43

3 3.6. The Autocovariance-Generating Function Invertibility 64 APPENDIX 3. A. Convergence Results for Infinite-Order Moving Average Processes 69 Exercises 70 References 71 4 Forecasting Principles of Forecasting Forecasts Based on an Infinite Number of Observations Forecasts Based on a Finite Number of Observations The Triangular Factorization of a Positive Definite Symmetric Matrix Updating a Linear Projection Optimal Forecasts for Gaussian Processes Sums of ARMA Processes Wold's Decomposition and the Box-Jenkins Modeling Philosophy 108 APPENDIX 4.A. Parallel Between OLS Regression and Linear Projection 113 APPENDIX 4.B. Triangular Factorization of the Covariance Matrix for an MA(1) Process 114 Exercises 115 References 116 Maximum Likelihood Estimation Introduction The Likelihood Function for a Gaussian AR(1) Process The Likelihood Function for a Gaussian AR(p) Process The Likelihood Function for a Gaussian AL4(1) Process The Likelihood Function for a Gaussian MA(q) Process The Likelihood Function for a Gaussian ARMA(p, q) Process Numerical Optimization 133 vi Contents

4 5.8. Statistical Inference with Maximum Likelihood Estimation Inequality Constraints 146 APPENDIX 5. A. Proofs of Chapter 5 Propositions 148 Exercises 150 References Spectral Analysis The Population Spectrum The Sample Periodogram Estimating the Population Spectrum Uses of Spectral Analysis 167 APPENDIX 6. A. Proofs of Chapter 6 Propositions 172 Exercises 178 References Asymptotic Distribution Theory Review of Asymptotic Distribution Theory Limit Theorems for Serially Dependent Observations 186 APPENDIX 7. A. Proofs of Chapter 7 Propositions 195 Exercises 198 References Linear Regression Models 8.1. Review of Ordinary Least Squares with Deterministic Regressors and i.i.d. Gaussian Disturbances Ordinary Least Squares Under More General Conditions Generalized Least Squares APPENDIX 8. A. Proofs of Chapter 8 Propositions 228 Exercises 230 References Linear Systems of Simultaneous Equations Simultaneous Equations Bias Instrumental Variables and Two-Stage Least Squares 238 Contents vii

5 9.3. Identification Full-Information Maximum Likelihood Estimation Estimation Based on the Reduced Form Overview of Simultaneous Equations Bias 252 APPENDIX 9. A. Proofs of Chapter 9 Proposition 253 Exercise 255 References Covariance-Stationary Vector Processes Introduction to Vector Autoregressions Autocovariances and Convergence Results for Vector Processes The Autocovariance-Generating Function for Vector Processes The Spectrum for Vector Processes The Sample Mean of a Vector Process APPENDIX 10. A. Proofs of Chapter 10 Propositions 285 Exercises 290 References Vector Autoregressions Maximum Likelihood Estimation and Hypothesis Testing for an Unrestricted Vector Autoregression Bivariate Granger Causality Tests Maximum Likelihood Estimation of Restricted Vector Autoregressions The Impulse-Response Function Variance Decomposition Vector Autoregressions and Structural Econometric Models Standard Errors for Impulse-Response Functions 336 APPENDIX ll.a. Proofs of Chapter 11 Propositions 340 APPENDIX ll.b. Calculation of Analytic Derivatives 344 Exercises 348 References 349 viii Contents

6 12 Bayesian Analysis Introduction to Bayesian Analysis Bayesian Analysis of Vector Autoregressions Numerical Bayesian Methods 362 APPENDIX 12. A. Proofs of Chapter 12 Propositions 366 Exercise 370 References The Kalman Filter The State-Space Representation of a Dynamic System Derivation of the Kalman Filter Forecasts Based on the State-Space Representation Maximum Likelihood Estimation of Parameters The Steady-State Kalman Filter Smoothing Statistical Inference with the Kalman Filter Time-Varying Parameters 399 APPENDIX 13. A. Proofs of Chapter 13 Propositions 403 Exercises 406 References Generalized Method of Moments Estimation by the Generalized Method of Moments Examples Extensions GMM and Maximum Likelihood Estimation 427 APPENDIX 14. A. Proofs of Chapter 14 Propositions 431 Exercise 432 References Models of Nonstationary Time Series Introduction Why Linear Time Trends and Unit Roots? Contents ix

7 15.3. Comparison of Trend-Stationary and Unit Root Processes The Meaning of Tests for Unit Roots Other Approaches to Trended Time Series 447 APPENDIX 15.A. Derivation of Selected Equations for Chapter References Processes with Deterministic Time Trends Asymptotic Distribution of OLS Estimates of the Simple Time Trend Model Hypothesis Testing for the Simple Time Trend Model Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend 463 APPENDIX 16. A. Derivation of Selected Equations for Chapter Exercises 474 References 474 Yl Univariate Processes with Unit Roots Introduction Brownian Motion The Functional Central Limit Theorem Asymptotic Properties of a First-Order Autoregression when the True Coefficient Is Unity Asymptotic Results for Unit Root Processes with General Serial Correlation Phillips-Perron Tests for Unit Roots Asymptotic Properties of a pth-order Autoregression and the Augmented Dickey-Fuller Tests for Unit Roots Other Approaches to Testing for Unit Roots Bayesian Analysis and Unit Roots APPENDIX 17.A. Proofs of Chapter 17 Propositions 534 Exercises 537 References 541 X Contents

8 18 Unit Roots in Multivariate Time Series Asymptotic Results for Nonstationary Vector Processes Vector Autoregressions Containing Unit Roots Spurious Regressions 557 APPENDIX 18. A. Proofs of Chapter 18 Propositions 562 Exercises 568 References Cointegration Introduction Testing the Null Hypothesis of No Cointegration Testing Hypotheses About the Cointegrating Vector 601 APPENDIX 19. A. Proofs of Chapter 19 Propositions 618 Exercises 625 References Full-Information Maximum Likelihood Analysis of Cointegrated Systems Canonical Correlation Maximum Likelihood Estimation Hypothesis Testing Overview of Unit Roots To Difference or Not to Difference? APPENDIX 20. A. Proofs of Chapter 20 Propositions 653 Exercises 655 References Time Series Models of Heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH) Extensions 665 APPENDIX 21. A. Derivation of Selected Equations for Chapter References 674 Contents xi

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