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