15.5. Other Approaches to Trended Time Series

Chapters in this book

1. Frontmatter i
2. Contents v
3. Preface xiii
4. 1 Difference Equations 1
5. 1.1. First-Order Difference Equations 1
6. 1.2. pth-Order Difference Equations 7
7. APPENDIX I.A. Proofs of Chapter 1 Propositions 21
8. Chapter 1 References 24
9. 2 Lag Operators 25
10. 2.1. Introduction 25
11. 2.2. First-Order Difference Equations 27
12. 2.3. Second-Order Difference Equations 29
13. 2.4. pth-Order Difference Equations 33
14. 2.5. Initial Conditions and Unbounded Sequences 36
15. Chapter 2 References 42
16. 3 Stationary ARMA Processes 43
17. 3.1. Expectations, Stationarity, and Ergodicity 43
18. 3.2. White Noise 47
19. 3.3. Moving Average Processes 48
20. 3.4. Autoregressive Processes 53
21. 3.5. Mixed Autoregressive Moving Average Processes 59
22. 3.6. The Autocovariance-Generating Function 61
23. 3.7. Invertibility 64
24. APPENDIX 3.A. Convergence Results for Infinite-Order Moving Average Processes 69
25. Chapter 3 Exercises 70
26. Chapter 3 References 71
27. 4 Forecasting 72
28. 4.1. Principles of Forecasting 72
29. 4.2. Forecasts Based on an Infinite Number of Observations 77
30. 4.3. Forecasts Based on a Finite Number of Observations 85
31. 4.4. The Triangular Factorization of a Positive Definite Symmetric Matrix 87
32. 4.5. Updating a Linear Projection 92
33. 4.6. Optimal Forecasts for Gaussian Processes 100
34. 4.7. Sums of ARM A Processes 102
35. 4.8. Wold's Decomposition and the Box-Jenkins Modeling Philosophy 108
36. APPENDIX 4.A. Parallel Between OLS Regression and Linear Projection 113
37. APPENDIX 4.B. Triangular Factorization of the Covariance Matrix for an MA(1) Process 114
38. Chapter 4 Exercises 115
39. Chapter 4 References 116
40. 5 Maximum Likelihood Estimation 117
41. 5.1. Introduction 117
42. 5.2. The Likelihood Function for a Gaussian AR(7J Process 118
43. 5.3. The Likelihood Function for a Gaussian AR(p) Process 123
44. 5.4. The Likelihood Function for a Gaussian MA(1) Process 127
45. 5.5. The Likelihood Function for a Gaussian MA(q) Process 130
46. 5.6. The Likelihood Function for a Gaussian ARMA(p, q) Process 132
47. 5.7. Numerical Optimization 133
48. 5.8. Statistical Inference with Maximum Likelihood Estimation 142
49. 5.9. Inequality Constraints 146
50. APPENDIX 5. A. Proofs of Chapter 5 Propositions 148
51. Chapter 5 Exercises 150
52. Chapter 5 References 150
53. 6 Spectral Analysis 152
54. 6.1. The Population Spectrum 152
55. 6.2. The Sample Periodogram 158
56. 6.3. Estimating the Population Spectrum 163
57. 6.4. Uses of Spectral Analysis 167
58. APPENDIX 6. A. Proofs of Chapter 6 Propositions 172
59. Chapter 6 Exercises 178
60. Chapter 6 References 178
61. 7 Asymptotic Distribution Theory 180
62. 7.1. Review of Asymptotic Distribution Theory 180
63. 7.2. Limit Theorems for Serially Dependent Observations 186
64. APPENDIX 7.A. Proofs of Chapter 7 Propositions 195
65. Chapter 7 Exercises 198
66. Chapter 7 Exercises 199
67. 8 Linear Regression Models 200
68. 8.1. Review of Ordinary Least Squares with Deterministic Regressors and i.i.d. Gaussian Disturbances 200
69. 8.2. Ordinary Least Squares Under More General Conditions 207
70. 8.3. Generalized Least Squares 220
71. APPENDIX 8. A. Proofs of Chapter 8 Propositions 228
72. Chapter 8 Exercises 230
73. Chapter 8 References 231
74. 9 Linear Systems of Simultaneous Equations 233
75. 9.1. Simultaneous Equations Bias 233
76. 9.2. Instrumental Variables and Two-Stage Least Squares 238
77. 9.3. Identification 243
78. 9.4. Full-Information Maximum Likelihood Estimation 247
79. 9.5 Estimation Based on the Reduced Form 250
80. 9.6. Overview of Simultaneous Equations Bias 252
81. APPENDIX 9.A. Proofs of Chapter 9 Proposition 253
82. Chapter 9 Exercise 255
83. Chapter 9 References 256
84. 10 Covariance-Stationary Vector Processes 257
85. 10.1. Introduction to Vector Autoregressions 257
86. 10.2. Autocovariances and Convergence Results for Vector Processes 261
87. 10.3. The Autocovariance-Generating Function for Vector Processes 266
88. 10.4. The Spectrum for Vector Processes 268
89. 10.5. The Sample Mean of a Vector Process 279
90. APPENDIX 10.A. Proofs of Chapter 10 Propositions 285
91. Chapter 10 Exercises 290
92. Chapter 10 References 290
93. 11 Vector Autoregressions 291
94. 11.1. Maximum Likelihood Estimation and Hypothesis Testing for an Unrestricted Vector Autoregression 291
95. 11.2. Bivariate Granger Causality Tests 302
96. 11.3. Maximum Likelihood Estimation of Restricted Vector Autoregressions 309
97. 11.4. The Impulse-Response Function 318
98. 11.5. Variance Decomposition 323
99. 11.6. Vector Autoregressions and Structural Econometric Models 324
100. 11.7. Standard Errors for Impulse-Response Functions 336
101. APPENDIX 11. A. Proofs of Chapter 11 Propositions 340
102. APPENDIX 11.B. Calculation of Analytic Derivatives 344
103. Chapter 11 Exercises 348
104. Chapter 11 References 349
105. 12 Bayesian Analysis 351
106. 12.1. Introduction to Bayesian Analysis 351
107. 12.2. Bayesian Analysis of Vector Autoregressions 360
108. 12.3. Numerical Bayesian Methods 362
109. APPENDIX 12.A. Proofs of Chapter 12 Propositions 366
110. Chapter 12 Exercise 370
111. Chapter 12 References 370
112. 13 The Kalman Filter 372
113. 13.1. The State-Space Representation of a Dynamic System 372
114. 13.2. Derivation of the Kalman Filter 377
115. 13.3. Forecasts Based on the State-Space Representation 381
116. 13.4. Maximum Likelihood Estimation 385
117. 13.5. The Steady-State Kalman Filter 389
118. 13.6. Smoothing 394
119. 13.7. Statistical Inference with the Kalman Filter 397
120. 13.8. Time-Varying Parameters 399
121. APPENDIX 13. A. Proofs of Chapter 13 Propositions 403
122. Chapter 13 Exercises 406
123. Chapter 13 References 407
124. 14 Generalized Method of Moments 409
125. 14.1. Estimation by the Generalized Method of Moments 409
126. 14.2. Examples 415
127. 14.3. Extensions 424
128. 14.4. GMM and Maximum Likelihood Estimation 427
129. APPENDIX 14. A. Proof of Chapter 14 Proposition 431
130. Chapter 14 Exercise 432
131. Chapter 14 References 433
132. 15 Models of Nonstationary Time Series 435
133. 15.1. Introduction 435
134. 15.2. Why Linear Time Trends and Unit Roots? 438
135. 15.3. Comparison of Trend-Stationary and Unit Root Processes 438
136. 15.4. The Meaning of Tests for Unit Roots 444
137. 15.5. Other Approaches to Trended Time Series 447
138. APPENDIX 15. A. Derivation of Selected Equations for Chapter 15 451
139. Chapter 15 References 452
140. 16 Processes with Deterministic Time Trends 454
141. 16.1. Asymptotic Distribution of OLS Estimates of the Simple Time Trend Model 454
142. 16.2. Hypothesis Testing for the Simple Time Trend Model 461
143. 16.3. Asymptotic Inference for an Autoregressive Process Around a Deterministic Time Trend 463
144. APPENDIX 16. A. Derivation of Selected Equations for Chapter 16 472
145. Chapter 16 Exercises 474
146. Chapter 16 References 474
147. 17 Univariate Processes with Unit Roots 475
148. 17.1. Introduction 475
149. 17.2. Brownian Motion 477
150. 17.3. The Functional Central Limit Theorem 479
151. 17.4. Asymptotic Properties of a First-Order Autoregression when the True Coefficient Is Unity 486
152. 17.5. Asymptotic Results for Unit Root Processes with General Serial Correlation 504
153. 17.6. Phillips-Perron Tests for Unit Roots 506
154. 17.7. Asymptotic Properties of a pth-Order Autoregression and the Augmented Dickey-Fuller Tests for Unit Roots 516
155. 17.8. Other Approaches to Testing for Unit Roots 531
156. 17.9. Bayesian Analysis and Unit Roots 532
157. APPENDIX 17.A. Proofs of Chapter 17 Propositions 534
158. Chapter 17 Exercises 537
159. Chapter 17 References 541
160. 18 Unit Roots in Multivariate Time Series 544
161. 18.1. Asymptotic Results for Nonstationary Vector Processes 544
162. 18.2. Vector Autoregressions Containing Unit Roots 549
163. 18.3. Spurious Regressions 557
164. APPENDIX 18.A. Proofs of Chapter 18 Propositions 562
165. Chapter 18 Exercises 568
166. Chapter 18 References 569
167. 19 Cointegration 571
168. 19.1. Introduction 571
169. 19.2. Testing the Null Hypothesis 582
170. 19.3. Testing Hypotheses About the Cointegrating Vector 601
171. APPENDIX 19. A. Proofs of Chapter 19 Propositions 618
172. Chapter 19 Exercises 625
173. Chapter 19 References 627
174. 20 Full-Information Maximum Likelihood Analysis of Cointegrated Systems 630
175. 20.1. Canonical Correlation 630
176. 20.2. Maximum Likelihood Estimation 635
177. 20.3. Hypothesis Testing 645
178. 20.4. Overview of Unit Roots—To Difference or Not to Difference? 651
179. APPENDIX 20.A. Proof of Chapter 20 Proposition 653
180. Chapter 20 Exercises 655
181. Chapter 20 References 655
182. 21 Time Series Models of Heteroskedasticity 657
183. 21.1. Autoregressive Conditional Heteroskedasticity (ARCH) 657
184. 21.2. Extensions 665
185. APPENDIX 21. A. Derivation of Selected Equations for Chapter 21 672
186. Chapter 21 References 674
187. 22 Modeling Time Series with Changes in Regime 677
188. 22.1. Introduction 677
189. 22.2. Markov Chains 678
190. 22.3. Statistical Analysis of i.i.d. Mixture Distributions 685
191. 22.4. Time Series Models of Changes in Regime 690
192. APPENDIX 22. A. Derivation of Selected Equations for Chapter 22 699
193. Chapter 22 Exercise 702
194. Chapter 22 Reference 702
195. A Mathematical Review 704
196. A.1. Trigonometry 704
197. A.2. Complex Numbers 708
198. A.3. Calculus 711
199. A.4. Matrix Algebra 721
200. A.5. Probability and Statistics 739
201. Appendix A References 750
202. B Statistical Tables 751
203. C Answers to Selected Exercises 769
204. D Greek Letters and Mathematical Symbols Used in the Text 786
205. Author Index 789
206. Subject Index 792