Time Series Analysis

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