Princeton University Press
Time Series Analysis
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and
About this book
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.
Author / Editor information
Reviews
“A carefully prepared and well written book. . . . Without doubt, it can be recommended as a very valuable encyclopedia and textbook for a reader who is looking for a mainly theoretical textbook which combines traditional time series analysis with a review of recent research areas.”—Journal of Economics
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Frontmatter
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Contents
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Preface
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1 Difference Equations
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1.1. First-Order Difference Equations
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1.2. pth-Order Difference Equations
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APPENDIX I.A. Proofs of Chapter 1 Propositions
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Chapter 1 References
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2 Lag Operators
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2.1. Introduction
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2.2. First-Order Difference Equations
27 -
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2.3. Second-Order Difference Equations
29 -
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2.4. pth-Order Difference Equations
33 -
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2.5. Initial Conditions and Unbounded Sequences
36 -
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Chapter 2 References
42 -
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3 Stationary ARMA Processes
43 -
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3.1. Expectations, Stationarity, and Ergodicity
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3.2. White Noise
47 -
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3.3. Moving Average Processes
48 -
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3.4. Autoregressive Processes
53 -
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3.5. Mixed Autoregressive Moving Average Processes
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3.6. The Autocovariance-Generating Function
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3.7. Invertibility
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APPENDIX 3.A. Convergence Results for Infinite-Order Moving Average Processes
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Chapter 3 Exercises
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Chapter 3 References
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4 Forecasting
72 -
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4.1. Principles of Forecasting
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4.2. Forecasts Based on an Infinite Number of Observations
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4.3. Forecasts Based on a Finite Number of Observations
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4.4. The Triangular Factorization of a Positive Definite Symmetric Matrix
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4.5. Updating a Linear Projection
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4.6. Optimal Forecasts for Gaussian Processes
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4.7. Sums of ARM A Processes
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4.8. Wold's Decomposition and the Box-Jenkins Modeling Philosophy
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APPENDIX 4.A. Parallel Between OLS Regression and Linear Projection
113 -
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APPENDIX 4.B. Triangular Factorization of the Covariance Matrix for an MA(1) Process
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Chapter 4 Exercises
115 -
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Chapter 4 References
116 -
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5 Maximum Likelihood Estimation
117 -
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5.1. Introduction
117 -
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5.2. The Likelihood Function for a Gaussian AR(7J Process
118 -
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5.3. The Likelihood Function for a Gaussian AR(p) Process
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5.4. The Likelihood Function for a Gaussian MA(1) Process
127 -
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5.5. The Likelihood Function for a Gaussian MA(q) Process
130 -
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5.6. The Likelihood Function for a Gaussian ARMA(p, q) Process
132 -
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5.7. Numerical Optimization
133 -
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5.8. Statistical Inference with Maximum Likelihood Estimation
142 -
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5.9. Inequality Constraints
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APPENDIX 5. A. Proofs of Chapter 5 Propositions
148 -
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Chapter 5 Exercises
150 -
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Chapter 5 References
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6 Spectral Analysis
152 -
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6.1. The Population Spectrum
152 -
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6.2. The Sample Periodogram
158 -
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6.3. Estimating the Population Spectrum
163 -
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6.4. Uses of Spectral Analysis
167 -
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APPENDIX 6. A. Proofs of Chapter 6 Propositions
172 -
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Chapter 6 Exercises
178 -
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Chapter 6 References
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7 Asymptotic Distribution Theory
180 -
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7.1. Review of Asymptotic Distribution Theory
180 -
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7.2. Limit Theorems for Serially Dependent Observations
186 -
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APPENDIX 7.A. Proofs of Chapter 7 Propositions
195 -
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Chapter 7 Exercises
198 -
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Chapter 7 Exercises
199 -
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8 Linear Regression Models
200 -
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8.1. Review of Ordinary Least Squares with Deterministic Regressors and i.i.d. Gaussian Disturbances
200 -
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8.2. Ordinary Least Squares Under More General Conditions
207 -
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8.3. Generalized Least Squares
220 -
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APPENDIX 8. A. Proofs of Chapter 8 Propositions
228 -
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Chapter 8 Exercises
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Chapter 8 References
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9 Linear Systems of Simultaneous Equations
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9.1. Simultaneous Equations Bias
233 -
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9.2. Instrumental Variables and Two-Stage Least Squares
238 -
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9.3. Identification
243 -
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9.4. Full-Information Maximum Likelihood Estimation
247 -
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9.5 Estimation Based on the Reduced Form
250 -
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9.6. Overview of Simultaneous Equations Bias
252 -
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APPENDIX 9.A. Proofs of Chapter 9 Proposition
253 -
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Chapter 9 Exercise
255 -
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Chapter 9 References
256 -
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10 Covariance-Stationary Vector Processes
257 -
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10.1. Introduction to Vector Autoregressions
257 -
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10.2. Autocovariances and Convergence Results for Vector Processes
261 -
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10.3. The Autocovariance-Generating Function for Vector Processes
266 -
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10.4. The Spectrum for Vector Processes
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10.5. The Sample Mean of a Vector Process
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APPENDIX 10.A. Proofs of Chapter 10 Propositions
285 -
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Chapter 10 Exercises
290 -
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Chapter 10 References
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11 Vector Autoregressions
291 -
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11.1. Maximum Likelihood Estimation and Hypothesis Testing for an Unrestricted Vector Autoregression
291 -
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11.2. Bivariate Granger Causality Tests
302 -
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11.3. Maximum Likelihood Estimation of Restricted Vector Autoregressions
309 -
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11.4. The Impulse-Response Function
318 -
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11.5. Variance Decomposition
323 -
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11.6. Vector Autoregressions and Structural Econometric Models
324 -
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11.7. Standard Errors for Impulse-Response Functions
336 -
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APPENDIX 11. A. Proofs of Chapter 11 Propositions
340 -
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APPENDIX 11.B. Calculation of Analytic Derivatives
344 -
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Chapter 11 Exercises
348 -
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Chapter 11 References
349 -
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12 Bayesian Analysis
351 -
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12.1. Introduction to Bayesian Analysis
351 -
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12.2. Bayesian Analysis of Vector Autoregressions
360 -
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12.3. Numerical Bayesian Methods
362 -
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APPENDIX 12.A. Proofs of Chapter 12 Propositions
366 -
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Chapter 12 Exercise
370 -
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Chapter 12 References
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13 The Kalman Filter
372 -
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13.1. The State-Space Representation of a Dynamic System
372 -
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13.2. Derivation of the Kalman Filter
377 -
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13.3. Forecasts Based on the State-Space Representation
381 -
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13.4. Maximum Likelihood Estimation
385 -
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13.5. The Steady-State Kalman Filter
389 -
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13.6. Smoothing
394 -
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13.7. Statistical Inference with the Kalman Filter
397 -
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13.8. Time-Varying Parameters
399 -
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APPENDIX 13. A. Proofs of Chapter 13 Propositions
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Chapter 13 Exercises
406 -
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Chapter 13 References
407 -
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14 Generalized Method of Moments
409 -
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14.1. Estimation by the Generalized Method of Moments
409 -
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14.2. Examples
415 -
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14.3. Extensions
424 -
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14.4. GMM and Maximum Likelihood Estimation
427 -
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APPENDIX 14. A. Proof of Chapter 14 Proposition
431 -
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Chapter 14 Exercise
432 -
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Chapter 14 References
433 -
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15 Models of Nonstationary Time Series
435 -
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15.1. Introduction
435 -
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15.2. Why Linear Time Trends and Unit Roots?
438 -
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15.3. Comparison of Trend-Stationary and Unit Root Processes
438 -
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15.4. The Meaning of Tests for Unit Roots
444 -
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15.5. Other Approaches to Trended Time Series
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APPENDIX 15. A. Derivation of Selected Equations for Chapter 15
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Chapter 15 References
452 -
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16 Processes with Deterministic Time Trends
454 -
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16.1. Asymptotic Distribution of OLS Estimates of the Simple Time Trend Model
454 -
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16.2. Hypothesis Testing for the Simple Time Trend Model
461 -
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16.3. Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend
463 -
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APPENDIX 16. A. Derivation of Selected Equations for Chapter 16
472 -
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Chapter 16 Exercises
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Chapter 16 References
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17 Univariate Processes with Unit Roots
475 -
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17.1. Introduction
475 -
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17.2. Brownian Motion
477 -
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17.3. The Functional Central Limit Theorem
479 -
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17.4. Asymptotic Properties of a First-Order Autoregression when the True Coefficient Is Unity
486 -
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17.5. Asymptotic Results for Unit Root Processes with General Serial Correlation
504 -
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17.6. Phillips-Perron Tests for Unit Roots
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17.7. Asymptotic Properties of a pth-Order Autoregression and the Augmented Dickey-Fuller Tests for Unit Roots
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17.8. Other Approaches to Testing for Unit Roots
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17.9. Bayesian Analysis and Unit Roots
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APPENDIX 17.A. Proofs of Chapter 17 Propositions
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Chapter 17 Exercises
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Chapter 17 References
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18 Unit Roots in Multivariate Time Series
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18.1. Asymptotic Results for Nonstationary Vector Processes
544 -
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18.2. Vector Autoregressions Containing Unit Roots
549 -
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18.3. Spurious Regressions
557 -
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APPENDIX 18.A. Proofs of Chapter 18 Propositions
562 -
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Chapter 18 Exercises
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Chapter 18 References
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19 Cointegration
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19.1. Introduction
571 -
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19.2. Testing the Null Hypothesis
582 -
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19.3. Testing Hypotheses About the Cointegrating Vector
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APPENDIX 19. A. Proofs of Chapter 19 Propositions
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Chapter 19 Exercises
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Chapter 19 References
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20 Full-Information Maximum Likelihood Analysis of Cointegrated Systems
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20.1. Canonical Correlation
630 -
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20.2. Maximum Likelihood Estimation
635 -
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20.3. Hypothesis Testing
645 -
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20.4. Overview of Unit Roots—To Difference or Not to Difference?
651 -
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APPENDIX 20.A. Proof of Chapter 20 Proposition
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Chapter 20 Exercises
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Chapter 20 References
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21 Time Series Models of Heteroskedasticity
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21.1. Autoregressive Conditional Heteroskedasticity (ARCH)
657 -
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21.2. Extensions
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APPENDIX 21. A. Derivation of Selected Equations for Chapter 21
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Chapter 21 References
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22 Modeling Time Series with Changes in Regime
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22.1. Introduction
677 -
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22.2. Markov Chains
678 -
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22.3. Statistical Analysis of i.i.d. Mixture Distributions
685 -
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22.4. Time Series Models of Changes in Regime
690 -
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APPENDIX 22. A. Derivation of Selected Equations for Chapter 22
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Chapter 22 Exercise
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Chapter 22 Reference
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A Mathematical Review
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A.1. Trigonometry
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A.2. Complex Numbers
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A.3. Calculus
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A.4. Matrix Algebra
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A.5. Probability and Statistics
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Appendix A References
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B Statistical Tables
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C Answers to Selected Exercises
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D Greek Letters and Mathematical Symbols Used in the Text
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Author Index
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Subject Index
792