## Unit Roots, Cointegration, and Structural ChangeTime series analysis has undergone many changes in recent years with the advent of unit roots and cointegration. Maddala and Kim present a comprehensive review of these important developments and examine structural change. The volume provides an analysis of unit root tests, problems with unit root testing, estimation of cointegration systems, cointegration tests, and econometric estimation with integrated regressors. The authors also present the Bayesian approach to these problems and bootstrap methods for small-sample inference. The chapters on structural change discuss the problems of unit root tests and cointegration under structural change, outliers and robust methods, the Markov-switching model and Harvey's structural time series model. Unit Roots, Cointegration and Structural Change is a major contribution to Themes in Modern Econometrics, of interest both to specialists and graduate and upper-undergraduate students. |

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### Contents

Basic concepts | 8 |

22 Some commonly used stationary models | 11 |

23 BoxJenkins methods | 17 |

24 Integrated variables and cointegration | 20 |

25 Spurious regression | 28 |

26 Deterministic trend and stochastic trend | 29 |

27 Detrending methods | 32 |

28 VAR ECM and ADL | 34 |

89 Bayesian inference on cointegrated systems | 287 |

810 Bayesian longrun prediction | 290 |

811 Conclusion | 291 |

References | 292 |

Fractional unit roots and fractional cointegration | 296 |

92 Unit root tests against fractional alternatives | 298 |

93 Estimation of ARFIMA models | 300 |

94 Estimation of fractionally cointegrated models | 302 |

29 Unit root tests | 37 |

210 Cointegration tests and ECM | 39 |

211 Summary | 41 |

References | 42 |

Unit roots and cointegration | 45 |

Unit roots | 47 |

32 Unit roots and Wiener processes | 49 |

33 Unit root tests without a deterministic trend | 60 |

34 DF test with a linear deterministic trend | 65 |

35 Specification of deterministic trends | 72 |

36 Unit root tests for a wide class of errors | 74 |

37 SarganBhargava and Bhargava tests | 82 |

38 Variance ratio tests | 86 |

39 Tests for TSP versus DSP | 87 |

310 Forecasting from TS versus DS models | 89 |

311 Summary and conclusions | 92 |

Issues in unit root testing | 98 |

42 Size distortion and low power of unit root tests | 100 |

43 Solutions to the problems of size and power | 103 |

MA roots | 116 |

45 Tests with stationarity as null | 120 |

46 Confirmatory analysis | 126 |

47 Frequency of observations and power of unit root tests | 129 |

48 Other types of nonstationarity | 131 |

49 Panel data unit root tests | 133 |

410 Uncertain unit roots and the pretesting problem | 139 |

411 Other unit root tests | 140 |

412 Medianunbiased estimation | 141 |

413 Summary and conclusions | 145 |

References | 146 |

Estimation of cointegrated systems | 155 |

EngleGranger methods | 156 |

54 A triangular system | 160 |

55 System estimation methods | 165 |

56 The identification problem | 173 |

57 Finite sample evidence | 175 |

58 Forecasting in cointegrated systems | 184 |

59 Miscellaneous other problems | 187 |

510 Summary and conclusions | 191 |

Tests for cointegration | 198 |

ECM tests | 203 |

64 Tests with cointegration as null | 205 |

65 Multiple equation methods | 211 |

66 Cointegration tests based on LCCA | 222 |

67 Other tests for cointegration | 226 |

68 Miscellaneous other problems | 228 |

69 Of what use are cointegration tests? | 233 |

610 Conclusions | 241 |

References | 242 |

Econometric modeling with integrated regressors | 249 |

72 I1 regressors cointegrated | 250 |

73 Unbalanced equations | 251 |

the ARDL model | 252 |

75 Uncertain unit roots | 254 |

76 Uncertain unit roots and cointegration | 256 |

77 Summary and conclusions | 258 |

Extensions of the basic model | 261 |

The Bayesian analysis of stochastic trends | 263 |

81 Introduction to Bayesian inference | 264 |

82 The posterior distribution of an autoregressive parameter | 266 |

83 Bayesian inference on the NelsonPlosser data | 268 |

84 The debate on the appropriate prior | 271 |

85 Classical tests versus Bayesian tests | 277 |

87 On testing point null hypotheses | 278 |

88 Further comments on prior distributions | 284 |

95 Empirical relevance of fractional unit roots | 303 |

96 Summary and conclusions | 305 |

References | 306 |

Small sample inference bootstrap methods | 309 |

103 The AR1 model | 322 |

104 Bootstrapping unit root tests | 325 |

105 The moving block bootstrap and extensions | 328 |

106 Issues in bootstrapping cointegrating regressions | 332 |

107 Miscellaneous other applications | 335 |

108 Conclusions | 336 |

Cointegrated systems with I2 variables | 342 |

112 Cointegration analysis with I2 and I1 variables | 348 |

113 Empirical applications | 355 |

114 Summary and conclusions | 358 |

359 | |

Seasonal unit roots and seasonal cointegration | 362 |

121 Effect of seasonal adjustment | 364 |

122 Seasonal integration | 365 |

123 Tests for seasonal unit roots | 366 |

124 The unobserved component model | 371 |

125 Seasonal cointegration | 375 |

126 Estimation of seasonally cointegrated systems | 376 |

127 Empirical evidence | 378 |

128 Periodic autoregression and periodic integration | 379 |

129 Periodic cointegration and seasonal cointegration | 381 |

1211 Conclusion | 382 |

383 | |

Structural change | 387 |

Structural change unit roots and cointegration | 389 |

131 Tests for structural change | 390 |

133 Tests with unknown break points | 391 |

134 A summary assessment | 398 |

135 Tests for unit roots under structural change | 399 |

136 The Bayesian approach | 402 |

137 A summary assessment of the empirical work | 407 |

138 Effect of structural change on cointegration tests | 410 |

139 Tests for structural change in cointegrated relationships | 411 |

1310 Miscellaneous other issues | 414 |

1311 Practical conclusions | 416 |

418 | |

Outliers and unit roots | 425 |

143 Effects of outliers on unit root tests | 428 |

144 Outlier detection | 437 |

145 Robust unit root tests | 440 |

146 Robust estimation of cointegrating regressions | 445 |

147 Outliers and seasonal unit roots | 448 |

449 | |

Regime switching models and structural time series models | 454 |

152 The Markov switching regression model | 455 |

153 The Hamilton model | 457 |

154 On the usefulness of the MSR model | 460 |

155 Extensions of the MSR model | 463 |

156 Gradual regime switching models | 466 |

157 A model with parameters following a random walk | 469 |

158 A general statespace model | 470 |

159 Derivation of the Kalman filter | 472 |

1510 Harveys structural time series model 1989 | 475 |

1511 Further comments on structural time series models | 477 |

1512 Summary and conclusions | 479 |

Future directions | 486 |

488 | |

A brief guide to asymptotic theory | 490 |

492 | |

500 | |

### Common terms and phrases

ADF test alternative applied argue asymptotic distribution autoregressive Bayesian analysis Bayesian inference bootstrap methods bootstrap-t break point chapter coefficients cointegrated systems cointegrating regression cointegrating relationships cointegrating vectors cointegration tests component consider critical values deterministic trend Dickey Dickey-Fuller Dickey-Fuller tests discussed drift dynamic Economics exchange rate F-statistics finite sample FM-OLS forecasting Granger inference integrated Johansen procedure Kalman filter least squares likelihood function long-run MA unit Maddala matrix Monte Carlo study MSR model nonstationary normal distribution null hypothesis OLS estimator outliers panel data paper parameters percent Perron Phillips problem pyt-i random walk regression model regressors residuals seasonal unit roots serial correlation series models single equation stationary stationary process stochastic trend structural breaks structural change suggested t-statistic test statistics tests for cointegration tests for unit tion unit root hypothesis unit root null unit root tests variables variance Wiener processes yt-i zero

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