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22.3. Statistical Analysis of i.i.d. Mixture Distributions

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Time Series Analysis
This chapter is in the book Time Series Analysis
Periodic Markov ChainsIf a Markov chain is irreducible, then there is one and only one eigenvalueequal to unity. However, there may be more than one eigenvalue on the unit circle,meaning that not all irreducible Markov chains are ergodic. For example, considera two-state Markov chain in which pXi = p22 = 0:p =The eigenvalues of this transition matrix are A! = 1 and A2 = -1, both of whichare on the unit circle. Thus, the matrix Pm does not converge to any fixed limit ofthe form wl' for this case. Instead, if the process is in state 1 at date t, thenit is certain to be there again for dates t + 2, t + 4, t + 6, . . . , with no ten-dency to converge as m —» °°. Such a Markov chain is said to be periodic withperiod 2.In general, it is possible to show that for any irreducible N-state Markovchain, all the eigenvalues of the transition matrix will be on or inside the unit circle.If there are K eigenvalues strictly on the unit circle with K > 1, then the chain issaid to be periodic with period K. Such chains have the property that the statescan be classified into K distinct classes, such that if the state at date t is from classa, then the state at date t + 1 is certain to be from class a + 1 (where class a + 1for a = K is interpreted to be class 1). Thus, there is a zero probability of returningto the original state st, and indeed zero probability of returning to any member ofthe original class a, except at horizons that are integer multiples of the period (suchas dates t + K, t + 2K, t + 3K, and so on). For further discussion of periodicMarkov chains, see Cox and Miller (1965).22.3. Statistical Analysis of i.i.d. Mixture DistributionsIn Section 22.4, we will consider autoregressive processes in which the parametersof the autoregression can change as the result of a regime-shift variable. The regimeitself will be described as the outcome of an unobserved Markov chain. Beforeanalyzing such processes, it is instructive first to consider a special case of theseprocesses known as i.i.d. mixture distributions.Let the regime that a given process is in at date t be indexed by an unobservedrandom variable st, where there are N possible regimes (s, = 1,2,..., or N).When the process is in regime 1, the observed variable y, is presumed to have beendrawn from a N(fi1, af) distribution. If the process is in regime 2, then y, is drawnfrom a N(fM2i a2) distribution, and so on. Hence, the density of yt conditional onthe random variable st taking on the value j is2a)for j = 1, 2, . . . , N. Here 6 is a vector of population parameters that includesMi, • • • , V-N and or2!, . . . ,a%.The unobserved regime {s,} is presumed to have been generated by someprobability distribution, for which the unconditional probability that st takes onthe value / is denoted TT,:P{s, = /; 6} = IT, for/ = 1, 2, . . . , N. [22.3.2]22.3. Statistical Analysis of i.i.d. Mixture Distributions 685
© 2020 Princeton University Press, Princeton

Periodic Markov ChainsIf a Markov chain is irreducible, then there is one and only one eigenvalueequal to unity. However, there may be more than one eigenvalue on the unit circle,meaning that not all irreducible Markov chains are ergodic. For example, considera two-state Markov chain in which pXi = p22 = 0:p =The eigenvalues of this transition matrix are A! = 1 and A2 = -1, both of whichare on the unit circle. Thus, the matrix Pm does not converge to any fixed limit ofthe form wl' for this case. Instead, if the process is in state 1 at date t, thenit is certain to be there again for dates t + 2, t + 4, t + 6, . . . , with no ten-dency to converge as m —» °°. Such a Markov chain is said to be periodic withperiod 2.In general, it is possible to show that for any irreducible N-state Markovchain, all the eigenvalues of the transition matrix will be on or inside the unit circle.If there are K eigenvalues strictly on the unit circle with K > 1, then the chain issaid to be periodic with period K. Such chains have the property that the statescan be classified into K distinct classes, such that if the state at date t is from classa, then the state at date t + 1 is certain to be from class a + 1 (where class a + 1for a = K is interpreted to be class 1). Thus, there is a zero probability of returningto the original state st, and indeed zero probability of returning to any member ofthe original class a, except at horizons that are integer multiples of the period (suchas dates t + K, t + 2K, t + 3K, and so on). For further discussion of periodicMarkov chains, see Cox and Miller (1965).22.3. Statistical Analysis of i.i.d. Mixture DistributionsIn Section 22.4, we will consider autoregressive processes in which the parametersof the autoregression can change as the result of a regime-shift variable. The regimeitself will be described as the outcome of an unobserved Markov chain. Beforeanalyzing such processes, it is instructive first to consider a special case of theseprocesses known as i.i.d. mixture distributions.Let the regime that a given process is in at date t be indexed by an unobservedrandom variable st, where there are N possible regimes (s, = 1,2,..., or N).When the process is in regime 1, the observed variable y, is presumed to have beendrawn from a N(fi1, af) distribution. If the process is in regime 2, then y, is drawnfrom a N(fM2i a2) distribution, and so on. Hence, the density of yt conditional onthe random variable st taking on the value j is2a)for j = 1, 2, . . . , N. Here 6 is a vector of population parameters that includesMi, • • • , V-N and or2!, . . . ,a%.The unobserved regime {s,} is presumed to have been generated by someprobability distribution, for which the unconditional probability that st takes onthe value / is denoted TT,:P{s, = /; 6} = IT, for/ = 1, 2, . . . , N. [22.3.2]22.3. Statistical Analysis of i.i.d. Mixture Distributions 685
© 2020 Princeton University Press, Princeton

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
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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
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