Time Series Analysis

James D. Hamilton

Book

Time Series Analysis

James D. Hamilton

Language: English

Published/Copyright: 1994

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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

James D. Hamilton is professor of economics at the University of California, San Diego.

Reviews

"I am extremely enthusiastic about this book. I think it will quickly become a classic. Like Sargent's and Varian's texts, it will be a centerpiece of the core cirriculum for graduate students."—John H. Cochrane, University of Chicago

“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

Topics

Business and Economics Political Economics Political Economics, other

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Preface Publicly Available Download PDF	xiii
1 Difference Equations Requires Authentication Unlicensed Licensed	1
1.1. First-Order Difference Equations Requires Authentication Unlicensed Licensed	1
1.2. pth-Order Difference Equations Requires Authentication Unlicensed Licensed	7
APPENDIX I.A. Proofs of Chapter 1 Propositions Requires Authentication Unlicensed Licensed	21
Chapter 1 References Requires Authentication Unlicensed Licensed	24
2 Lag Operators Requires Authentication Unlicensed Licensed	25
2.1. Introduction Requires Authentication Unlicensed Licensed	25
2.2. First-Order Difference Equations Requires Authentication Unlicensed Licensed	27
2.3. Second-Order Difference Equations Requires Authentication Unlicensed Licensed	29
2.4. pth-Order Difference Equations Requires Authentication Unlicensed Licensed	33
2.5. Initial Conditions and Unbounded Sequences Requires Authentication Unlicensed Licensed	36
Chapter 2 References Requires Authentication Unlicensed Licensed	42
3 Stationary ARMA Processes Requires Authentication Unlicensed Licensed	43
3.1. Expectations, Stationarity, and Ergodicity Requires Authentication Unlicensed Licensed	43
3.2. White Noise Requires Authentication Unlicensed Licensed	47
3.3. Moving Average Processes Requires Authentication Unlicensed Licensed	48
3.4. Autoregressive Processes Requires Authentication Unlicensed Licensed	53
3.5. Mixed Autoregressive Moving Average Processes Requires Authentication Unlicensed Licensed	59
3.6. The Autocovariance-Generating Function Requires Authentication Unlicensed Licensed	61
3.7. Invertibility Requires Authentication Unlicensed Licensed	64
APPENDIX 3.A. Convergence Results for Infinite-Order Moving Average Processes Requires Authentication Unlicensed Licensed	69
Chapter 3 Exercises Requires Authentication Unlicensed Licensed	70
Chapter 3 References Requires Authentication Unlicensed Licensed	71
4 Forecasting Requires Authentication Unlicensed Licensed	72
4.1. Principles of Forecasting Requires Authentication Unlicensed Licensed	72
4.2. Forecasts Based on an Infinite Number of Observations Requires Authentication Unlicensed Licensed	77
4.3. Forecasts Based on a Finite Number of Observations Requires Authentication Unlicensed Licensed	85
4.4. The Triangular Factorization of a Positive Definite Symmetric Matrix Requires Authentication Unlicensed Licensed	87
4.5. Updating a Linear Projection Requires Authentication Unlicensed Licensed	92
4.6. Optimal Forecasts for Gaussian Processes Requires Authentication Unlicensed Licensed	100
4.7. Sums of ARM A Processes Requires Authentication Unlicensed Licensed	102
4.8. Wold's Decomposition and the Box-Jenkins Modeling Philosophy Requires Authentication Unlicensed Licensed	108
APPENDIX 4.A. Parallel Between OLS Regression and Linear Projection Requires Authentication Unlicensed Licensed	113
APPENDIX 4.B. Triangular Factorization of the Covariance Matrix for an MA(1) Process Requires Authentication Unlicensed Licensed	114
Chapter 4 Exercises Requires Authentication Unlicensed Licensed	115
Chapter 4 References Requires Authentication Unlicensed Licensed	116
5 Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	117
5.1. Introduction Requires Authentication Unlicensed Licensed	117
5.2. The Likelihood Function for a Gaussian AR(7J Process Requires Authentication Unlicensed Licensed	118
5.3. The Likelihood Function for a Gaussian AR(p) Process Requires Authentication Unlicensed Licensed	123
5.4. The Likelihood Function for a Gaussian MA(1) Process Requires Authentication Unlicensed Licensed	127
5.5. The Likelihood Function for a Gaussian MA(q) Process Requires Authentication Unlicensed Licensed	130
5.6. The Likelihood Function for a Gaussian ARMA(p, q) Process Requires Authentication Unlicensed Licensed	132
5.7. Numerical Optimization Requires Authentication Unlicensed Licensed	133
5.8. Statistical Inference with Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	142
5.9. Inequality Constraints Requires Authentication Unlicensed Licensed	146
APPENDIX 5. A. Proofs of Chapter 5 Propositions Requires Authentication Unlicensed Licensed	148
Chapter 5 Exercises Requires Authentication Unlicensed Licensed	150
Chapter 5 References Requires Authentication Unlicensed Licensed	150
6 Spectral Analysis Requires Authentication Unlicensed Licensed	152
6.1. The Population Spectrum Requires Authentication Unlicensed Licensed	152
6.2. The Sample Periodogram Requires Authentication Unlicensed Licensed	158
6.3. Estimating the Population Spectrum Requires Authentication Unlicensed Licensed	163
6.4. Uses of Spectral Analysis Requires Authentication Unlicensed Licensed	167
APPENDIX 6. A. Proofs of Chapter 6 Propositions Requires Authentication Unlicensed Licensed	172
Chapter 6 Exercises Requires Authentication Unlicensed Licensed	178
Chapter 6 References Requires Authentication Unlicensed Licensed	178
7 Asymptotic Distribution Theory Requires Authentication Unlicensed Licensed	180
7.1. Review of Asymptotic Distribution Theory Requires Authentication Unlicensed Licensed	180
7.2. Limit Theorems for Serially Dependent Observations Requires Authentication Unlicensed Licensed	186
APPENDIX 7.A. Proofs of Chapter 7 Propositions Requires Authentication Unlicensed Licensed	195
Chapter 7 Exercises Requires Authentication Unlicensed Licensed	198
Chapter 7 Exercises Requires Authentication Unlicensed Licensed	199
8 Linear Regression Models Requires Authentication Unlicensed Licensed	200
8.1. Review of Ordinary Least Squares with Deterministic Regressors and i.i.d. Gaussian Disturbances Requires Authentication Unlicensed Licensed	200
8.2. Ordinary Least Squares Under More General Conditions Requires Authentication Unlicensed Licensed	207
8.3. Generalized Least Squares Requires Authentication Unlicensed Licensed	220
APPENDIX 8. A. Proofs of Chapter 8 Propositions Requires Authentication Unlicensed Licensed	228
Chapter 8 Exercises Requires Authentication Unlicensed Licensed	230
Chapter 8 References Requires Authentication Unlicensed Licensed	231
9 Linear Systems of Simultaneous Equations Requires Authentication Unlicensed Licensed	233
9.1. Simultaneous Equations Bias Requires Authentication Unlicensed Licensed	233
9.2. Instrumental Variables and Two-Stage Least Squares Requires Authentication Unlicensed Licensed	238
9.3. Identification Requires Authentication Unlicensed Licensed	243
9.4. Full-Information Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	247
9.5 Estimation Based on the Reduced Form Requires Authentication Unlicensed Licensed	250
9.6. Overview of Simultaneous Equations Bias Requires Authentication Unlicensed Licensed	252
APPENDIX 9.A. Proofs of Chapter 9 Proposition Requires Authentication Unlicensed Licensed	253
Chapter 9 Exercise Requires Authentication Unlicensed Licensed	255
Chapter 9 References Requires Authentication Unlicensed Licensed	256
10 Covariance-Stationary Vector Processes Requires Authentication Unlicensed Licensed	257
10.1. Introduction to Vector Autoregressions Requires Authentication Unlicensed Licensed	257
10.2. Autocovariances and Convergence Results for Vector Processes Requires Authentication Unlicensed Licensed	261
10.3. The Autocovariance-Generating Function for Vector Processes Requires Authentication Unlicensed Licensed	266
10.4. The Spectrum for Vector Processes Requires Authentication Unlicensed Licensed	268
10.5. The Sample Mean of a Vector Process Requires Authentication Unlicensed Licensed	279
APPENDIX 10.A. Proofs of Chapter 10 Propositions Requires Authentication Unlicensed Licensed	285
Chapter 10 Exercises Requires Authentication Unlicensed Licensed	290
Chapter 10 References Requires Authentication Unlicensed Licensed	290
11 Vector Autoregressions Requires Authentication Unlicensed Licensed	291
11.1. Maximum Likelihood Estimation and Hypothesis Testing for an Unrestricted Vector Autoregression Requires Authentication Unlicensed Licensed	291
11.2. Bivariate Granger Causality Tests Requires Authentication Unlicensed Licensed	302
11.3. Maximum Likelihood Estimation of Restricted Vector Autoregressions Requires Authentication Unlicensed Licensed	309
11.4. The Impulse-Response Function Requires Authentication Unlicensed Licensed	318
11.5. Variance Decomposition Requires Authentication Unlicensed Licensed	323
11.6. Vector Autoregressions and Structural Econometric Models Requires Authentication Unlicensed Licensed	324
11.7. Standard Errors for Impulse-Response Functions Requires Authentication Unlicensed Licensed	336
APPENDIX 11. A. Proofs of Chapter 11 Propositions Requires Authentication Unlicensed Licensed	340
APPENDIX 11.B. Calculation of Analytic Derivatives Requires Authentication Unlicensed Licensed	344
Chapter 11 Exercises Requires Authentication Unlicensed Licensed	348
Chapter 11 References Requires Authentication Unlicensed Licensed	349
12 Bayesian Analysis Requires Authentication Unlicensed Licensed	351
12.1. Introduction to Bayesian Analysis Requires Authentication Unlicensed Licensed	351
12.2. Bayesian Analysis of Vector Autoregressions Requires Authentication Unlicensed Licensed	360
12.3. Numerical Bayesian Methods Requires Authentication Unlicensed Licensed	362
APPENDIX 12.A. Proofs of Chapter 12 Propositions Requires Authentication Unlicensed Licensed	366
Chapter 12 Exercise Requires Authentication Unlicensed Licensed	370
Chapter 12 References Requires Authentication Unlicensed Licensed	370
13 The Kalman Filter Requires Authentication Unlicensed Licensed	372
13.1. The State-Space Representation of a Dynamic System Requires Authentication Unlicensed Licensed	372
13.2. Derivation of the Kalman Filter Requires Authentication Unlicensed Licensed	377
13.3. Forecasts Based on the State-Space Representation Requires Authentication Unlicensed Licensed	381
13.4. Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	385
13.5. The Steady-State Kalman Filter Requires Authentication Unlicensed Licensed	389
13.6. Smoothing Requires Authentication Unlicensed Licensed	394
13.7. Statistical Inference with the Kalman Filter Requires Authentication Unlicensed Licensed	397
13.8. Time-Varying Parameters Requires Authentication Unlicensed Licensed	399
APPENDIX 13. A. Proofs of Chapter 13 Propositions Requires Authentication Unlicensed Licensed	403
Chapter 13 Exercises Requires Authentication Unlicensed Licensed	406
Chapter 13 References Requires Authentication Unlicensed Licensed	407
14 Generalized Method of Moments Requires Authentication Unlicensed Licensed	409
14.1. Estimation by the Generalized Method of Moments Requires Authentication Unlicensed Licensed	409
14.2. Examples Requires Authentication Unlicensed Licensed	415
14.3. Extensions Requires Authentication Unlicensed Licensed	424
14.4. GMM and Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	427
APPENDIX 14. A. Proof of Chapter 14 Proposition Requires Authentication Unlicensed Licensed	431
Chapter 14 Exercise Requires Authentication Unlicensed Licensed	432
Chapter 14 References Requires Authentication Unlicensed Licensed	433
15 Models of Nonstationary Time Series Requires Authentication Unlicensed Licensed	435
15.1. Introduction Requires Authentication Unlicensed Licensed	435
15.2. Why Linear Time Trends and Unit Roots? Requires Authentication Unlicensed Licensed	438
15.3. Comparison of Trend-Stationary and Unit Root Processes Requires Authentication Unlicensed Licensed	438
15.4. The Meaning of Tests for Unit Roots Requires Authentication Unlicensed Licensed	444
15.5. Other Approaches to Trended Time Series Requires Authentication Unlicensed Licensed	447
APPENDIX 15. A. Derivation of Selected Equations for Chapter 15 Requires Authentication Unlicensed Licensed	451
Chapter 15 References Requires Authentication Unlicensed Licensed	452
16 Processes with Deterministic Time Trends Requires Authentication Unlicensed Licensed	454
16.1. Asymptotic Distribution of OLS Estimates of the Simple Time Trend Model Requires Authentication Unlicensed Licensed	454
16.2. Hypothesis Testing for the Simple Time Trend Model Requires Authentication Unlicensed Licensed	461
16.3. Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend Requires Authentication Unlicensed Licensed	463
APPENDIX 16. A. Derivation of Selected Equations for Chapter 16 Requires Authentication Unlicensed Licensed	472
Chapter 16 Exercises Requires Authentication Unlicensed Licensed	474
Chapter 16 References Requires Authentication Unlicensed Licensed	474
17 Univariate Processes with Unit Roots Requires Authentication Unlicensed Licensed	475
17.1. Introduction Requires Authentication Unlicensed Licensed	475
17.2. Brownian Motion Requires Authentication Unlicensed Licensed	477
17.3. The Functional Central Limit Theorem Requires Authentication Unlicensed Licensed	479
17.4. Asymptotic Properties of a First-Order Autoregression when the True Coefficient Is Unity Requires Authentication Unlicensed Licensed	486
17.5. Asymptotic Results for Unit Root Processes with General Serial Correlation Requires Authentication Unlicensed Licensed	504
17.6. Phillips-Perron Tests for Unit Roots Requires Authentication Unlicensed Licensed	506
17.7. Asymptotic Properties of a pth-Order Autoregression and the Augmented Dickey-Fuller Tests for Unit Roots Requires Authentication Unlicensed Licensed	516
17.8. Other Approaches to Testing for Unit Roots Requires Authentication Unlicensed Licensed	531
17.9. Bayesian Analysis and Unit Roots Requires Authentication Unlicensed Licensed	532
APPENDIX 17.A. Proofs of Chapter 17 Propositions Requires Authentication Unlicensed Licensed	534
Chapter 17 Exercises Requires Authentication Unlicensed Licensed	537
Chapter 17 References Requires Authentication Unlicensed Licensed	541
18 Unit Roots in Multivariate Time Series Requires Authentication Unlicensed Licensed	544
18.1. Asymptotic Results for Nonstationary Vector Processes Requires Authentication Unlicensed Licensed	544
18.2. Vector Autoregressions Containing Unit Roots Requires Authentication Unlicensed Licensed	549
18.3. Spurious Regressions Requires Authentication Unlicensed Licensed	557
APPENDIX 18.A. Proofs of Chapter 18 Propositions Requires Authentication Unlicensed Licensed	562
Chapter 18 Exercises Requires Authentication Unlicensed Licensed	568
Chapter 18 References Requires Authentication Unlicensed Licensed	569
19 Cointegration Requires Authentication Unlicensed Licensed	571
19.1. Introduction Requires Authentication Unlicensed Licensed	571
19.2. Testing the Null Hypothesis Requires Authentication Unlicensed Licensed	582
19.3. Testing Hypotheses About the Cointegrating Vector Requires Authentication Unlicensed Licensed	601
APPENDIX 19. A. Proofs of Chapter 19 Propositions Requires Authentication Unlicensed Licensed	618
Chapter 19 Exercises Requires Authentication Unlicensed Licensed	625
Chapter 19 References Requires Authentication Unlicensed Licensed	627
20 Full-Information Maximum Likelihood Analysis of Cointegrated Systems Requires Authentication Unlicensed Licensed	630
20.1. Canonical Correlation Requires Authentication Unlicensed Licensed	630
20.2. Maximum Likelihood Estimation Requires Authentication Unlicensed Licensed	635
20.3. Hypothesis Testing Requires Authentication Unlicensed Licensed	645
20.4. Overview of Unit Roots—To Difference or Not to Difference? Requires Authentication Unlicensed Licensed	651
APPENDIX 20.A. Proof of Chapter 20 Proposition Requires Authentication Unlicensed Licensed	653
Chapter 20 Exercises Requires Authentication Unlicensed Licensed	655
Chapter 20 References Requires Authentication Unlicensed Licensed	655
21 Time Series Models of Heteroskedasticity Requires Authentication Unlicensed Licensed	657
21.1. Autoregressive Conditional Heteroskedasticity (ARCH) Requires Authentication Unlicensed Licensed	657
21.2. Extensions Requires Authentication Unlicensed Licensed	665
APPENDIX 21. A. Derivation of Selected Equations for Chapter 21 Requires Authentication Unlicensed Licensed	672
Chapter 21 References Requires Authentication Unlicensed Licensed	674
22 Modeling Time Series with Changes in Regime Requires Authentication Unlicensed Licensed	677
22.1. Introduction Requires Authentication Unlicensed Licensed	677
22.2. Markov Chains Requires Authentication Unlicensed Licensed	678
22.3. Statistical Analysis of i.i.d. Mixture Distributions Requires Authentication Unlicensed Licensed	685
22.4. Time Series Models of Changes in Regime Requires Authentication Unlicensed Licensed	690
APPENDIX 22. A. Derivation of Selected Equations for Chapter 22 Requires Authentication Unlicensed Licensed	699
Chapter 22 Exercise Requires Authentication Unlicensed Licensed	702
Chapter 22 Reference Requires Authentication Unlicensed Licensed	702
A Mathematical Review Requires Authentication Unlicensed Licensed	704
A.1. Trigonometry Requires Authentication Unlicensed Licensed	704
A.2. Complex Numbers Requires Authentication Unlicensed Licensed	708
A.3. Calculus Requires Authentication Unlicensed Licensed	711
A.4. Matrix Algebra Requires Authentication Unlicensed Licensed	721
A.5. Probability and Statistics Requires Authentication Unlicensed Licensed	739
Appendix A References Requires Authentication Unlicensed Licensed	750
B Statistical Tables Requires Authentication Unlicensed Licensed	751
C Answers to Selected Exercises Requires Authentication Unlicensed Licensed	769
D Greek Letters and Mathematical Symbols Used in the Text Requires Authentication Unlicensed Licensed	786
Author Index Requires Authentication Unlicensed Licensed	789
Subject Index Requires Authentication Unlicensed Licensed	792

Publishing information

Pages and Images/Illustrations in book

eBook published on:

September 1, 2020

eBook ISBN:

9780691218632

Pages and Images/Illustrations in book

Main content:

816

https://doi.org/10.1515/9780691218632

eBook ISBN: 9780691218632

Keywords for this book

Estimation; Unit root; Variable (mathematics); Parameter; Time series; Null hypothesis; Likelihood function; Asymptotic distribution; Variance; Maximum likelihood estimation; Autoregressive model; Stationary process; Cointegration; Coefficient; Prior probability; Markov chain; Random variable; Autocorrelation; Equation; F-test; Vector autoregression; Likelihood-ratio test; Probability; Eigenvalues and eigenvectors; Estimator; Inference; Kalman filter; Projection (linear algebra); Standard error; Generalized method of moments; Dickey–Fuller test; Forecasting; Autocovariance; Simultaneous equations; Constant term; Existential quantification; Rate of convergence; Quasi-maximum likelihood estimate; Error term; Bayesian inference; Granger causality; Forecast error; Autoregressive conditional heteroskedasticity; Least squares; Statistical hypothesis testing; Student's t-test; Linear regression; Fisher information; Special case; Nuisance parameter; Newton's method; Expectation–maximization algorithm; Statistic; Chi-squared test; Mathematical optimization; Central limit theorem; Autoregressive–moving-average model; Dummy variable (statistics); Box–Jenkins; Unit root test; Order condition; Maximum a posteriori estimation; Standard deviation; Accuracy and precision; Errors and residuals; Canonical correlation; F-distribution; Covariance matrix; T-statistic; Confidence region

Audience(s) for this book

College/higher education;Professional and scholarly;

Time Series Analysis

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